frobenioids-1.dvi

THE GEOMETRY OF FROBENIOIDS I: THE GENERAL THEORY Shinichi Mochizuki June 2008 We develop the theory of Frobenioids, which may be regarded as a category-theoretic abstraction of the theory of divisors and line bundles on models of ﬁnite separable extensions of a given function ﬁeld or number ﬁeld. This sort of abstraction is analogous to the role of Galois categories in Galois theory or monoids in the geometry of log schemes. This abstract category-theoretic framework preserves many of the important features of the classical theory of divisors and line bundles on models of ﬁnite separable extensions of a function ﬁeld or number ﬁeld such as the global degree of an arithmetic line bundle over a number ﬁeld, but also exhibits interesting new phenomena, such as a “Frobenius endomorphism” of the Frobenioid associated to a number ﬁeld. Introduction §0. Notations and Conventions §1. Deﬁnitions and First Properties §2. Frobenius Functors §3. Category-theoreticity of the Base and Frobenius Degree §4. Category-theoreticity of the Divisor Monoid §5. Model Frobenioids §6. Some Motivating Examples Appendix: Slim Exponentiation Index Chart of Types of Morphisms in a Frobenioid Bibliography Introduction §I1. Technical Summary §I2. Abstract Combinatorialization of Arithmetic Geometry §I3. Frobenius Endomorphisms of a Number Field §I4. Étale-like vs. Frobenius-like Structures Acknowledgements 2000 Mathematical Subject Classiﬁcation: Primary 14A99; Secondary 11G99. Keywords and Phrases: categories; Galois categories; Frobenius; monoids; log schemes. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI §I1. Technical Summary In the present paper, we introduce the notion of a Frobenioid. The simplest kind of Frobenioid “F M ” is the non-commutative monoid given by forming the “semi-direct product monoid” of a given commutative monoid M with the multi- plicative monoid of positive integers N ≥1 [cf. §0], where n ∈ N ≥1 acts on M by multiplication by n; that is to say, the underlying set of F M is the product M × N ≥1 equipped with the monoid structure is given as follows: if a 1 , a 2 ∈ M , n 1 , n 2 ∈ N ≥1 , then (a 1 , n 1 ) · (a 2 , n 2 ) = (a 1 + n 1 · a 2 , n 1 · n 2 ) [cf. Deﬁnition 1.1, (iii)]. For instance, when M is taken to be the additive monoid of nonnegative integers Z ≥0 [cf. §0], def we shall write F = F Z ≥0 and refer to F as the standard Frobenioid. Note that in general, any monoid [such as F M , for instance] may be thought of as a category, i.e., the category with precisely one object whose monoid of endomorphisms is the given monoid. More generally, one may start with a “family of commutative monoids” Φ on a “base category” D [where Φ, D satisfy certain properties] and form the associated elementary Frobenioid F Φ by taking the “semi-direct product” of N ≥1 with Φ [cf. Deﬁnition 1.1, (iii), for more details]. Here, F Φ is a category. In general, a Frobenioid C is a category equipped with a functor C → F Φ to an elementary Frobenioid F Φ satisfying certain properties [cf. Deﬁnition 1.3 for more details] to the eﬀect that the structure of C is “substantially reﬂected” in this functor C → F Φ . From the point of view of conventional arithmetic geometry, a Frobenioid may be thought of as a sort of a category-theoretic abstraction of the theory of divisors and line bundles on models of ﬁnite separable extensions of a given function ﬁeld or number ﬁeld. That is to say, the base category D corresponds to the category of models of ﬁnite separable extensions of a given function ﬁeld or number ﬁeld; the functor Φ corresponds to the divisors on such models; the “N ≥1 portion” of F Φ corresponds to the operation of multiplying a divisor by an element n ∈ N ≥1 [or, if one considers the line bundle associated to such a divisor, to the operation of forming the n-th tensor power of the line bundle]. In some sense, the main result of the theory of present paper is the following: Under various technical conditions, the functor C → F Φ that determines the structure of C as a Frobenioid may be reconstructed purely category- theoretically, i.e., from the structure of C as a category [cf. Corollary 4.11]. These technical conditions are typically satisﬁed by Frobenioids that arise naturally from arithmetic geometry [cf. Theorems 6.2, 6.4]. Also, we observe that these technical conditions appear unlikely to be superﬂuous. Indeed, we also give various examples, involving Frobenioids which do not satisfy various of these technical conditions, of equivalences of categories with respect to which various portions of THE GEOMETRY OF FROBENIOIDS I 3 the functor C → F Φ are not preserved [cf. Examples 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 4.3]. Perhaps the most fundamental example of this phenomenon of “the intrinsic category-theoretic reconstruction of C → F Φ from C” is the following. The proto- type of a base category D is given by [the subcategory of connected objects of] a Galois category, i.e., a category in which the monoids of endomorphisms of objects have the structure of ﬁnite groups. On the other hand, the prototype of the “non- base category portion” of a Frobenioid, i.e., the “relative structure of C over D”, is given by the monoid “F” [or, more generally, the monoids “F M ”] discussed above. Then one central aspect of the phenomenon that “the relative structure of C over D is never confused with the structure of D” is illustrated by the following easily veriﬁed observation: If G is a ﬁnite group, then any homomorphism of monoids F → G factors through the natural surjection F N ≥1 . [We refer to Remark 3.1.2 for more details.] Note that this property fails to hold if, for instance, one replaces F = F Z ≥0 by Z ≥0 [and the surjection F N ≥1 by the surjection Z ≥0 {0}]. Put another way, this property may be thought of as a consequence of the non-abelian nature of F. In particular, if one thinks of the category-theoretic reconstructibility of the functor “C → F Φ ” as a sort of rigidity, then this property is vaguely reminiscent of the “extraordinary rigidity” asserted by Grothendieck in descriptions of his anabelian philosophy. After deﬁning and examining the ﬁrst properties of Frobenioids in §1, we pro- ceed to discuss, in §2, various versions of “Frobenius functors” on Frobenioids, which are intended as category-theoretic abstractions of the Frobenius morphism in positive characteristic algebraic geometry [cf. Remark 6.2.1]. In §3, we begin the category-theoretic reconstruction of the functor “C → F Φ ” by showing that, under certain conditions, the base category and “Frobenius degree” [i.e., in eﬀect, the “N ≥1 portion of F Φ ”] may be reconstructed category-theoretically [cf. Theorem 3.4]. In the theory of §3, we apply a certain purely category-theoretic technique, which we shall refer to as “slim exponentiation”; this technique is entirely independent of the theory of Frobenioids and is discussed in detail in the Appendix. In §4, we then complete the category-theoretic reconstruction of the functor “C → F Φ ” by show- ing that, under certain conditions, the divisor monoid Φ may also be reconstructed category-theoretically [cf. Theorem 4.9]. In §5, we study the extent to which, under certain conditions, one may write down “explicit models” of fairly general Frobenioids, in a fashion reminiscent of the explicit description of the elementary Frobenioid F Φ [cf. Theorem 5.2]. This study leads naturally to the investigation of various auxiliary structures on a Frobenioid, namely, base sections and base- Frobenius pairs, that may be used to relate a given Frobenioid satisfying certain conditions to such a “model Frobenioid” [cf. Theorem 5.2, (iv)]. One important technique in the theory of §3, §4, §5 is the operation of pass- ing from a Frobenioid to the perfection or realiﬁcation of the Frobenioid. Roughly speaking, from the point of view of the monoid F = F Z ≥0 introduced above, these 4 SHINICHI MOCHIZUKI operations correspond, respectively, to passing from “Z ≥0 ” to the monoids “Q ≥0 ” [in the case of the perfection] or “R ≥0 ” [in the case of the realiﬁcation]. Another important technique in this theory is the operation of passing to the birationaliza- tion of a Frobenioid. This may be thought of as a category-theoretic abstraction of the notion of “working with rational functions” in algebraic geometry; alternatively, from the point of view of the monoid F = F Z ≥0 introduced above, it may be thought of as corresponding to the operation of passing from “Z ≥0 ” to the groupiﬁcation “Z” of Z ≥0 . operation eﬀect on Z ≥0 ⊆ F perfection Z ≥0 Q ≥0 realiﬁcation Z ≥0 R ≥0 birationalization Z ≥0 Z Finally, in §6, we consider the main motivating examples of Frobenioids that arise from number ﬁelds and function ﬁelds. In particular, we observe in passing that this “Frobenioid-theoretic formulation of the elementary arithmetic of number ﬁelds” also gives rise to some interesting “Frobenioid-theoretic interpretations” of such classical results in number theory as the Dirichlet unit theorem and Tchebotarev’s density theorem, as well as a result in transcendence theory due to Lang [cf. Theo- rem 6.4, (i), (iii), (iv)]. §I2. Abstract Combinatorialization of Arithmetic Geometry From a somewhat more conceptual point of view, one central theme of the present paper is the goal of “abstract combinatorialization of scheme-theoretic arithmetic geometry”. Classical examples of this phenomenon of “abstract com- binatorialization” may be seen in the theory of Galois categories or the theory of monoids in the geometry of log schemes [or, more classically, toric varieties]. That is to say, even if one starts by considering various ﬁnite étale coverings of schemes, the associated Galois category is a purely “abstract combinatorial” mathematical object that captures the “Galois structure” of the various coverings involved in a fashion that is entirely independent of scheme theory. In a similar vein, although a monoid of the sort that appears in log geometry arises as a submonoid of the multiplicative monoid determined by some commutative ring, the “abstract com- binatorial” structure of such a monoid is suﬃcient to capture various essential properties [such as normality, etc.] of the ring structure of the ambient ring in a fashion that is entirely independent of ring/scheme theory. A somewhat less classical example of this phenomenon of “abstract combinato- rialization of scheme theory” is given by the theory of [Mzk8], where it is shown that very general locally noetherian log schemes may be “represented” by categories, in the sense that equivalences between such categories arise from uniquely determined isomorphisms of log schemes. The theory of [Mzk8] is generalized in [Mzk9] so as to take into account the archimedean primes of log schemes which are locally of ﬁnite THE GEOMETRY OF FROBENIOIDS I 5 type over a Zariski localization of [the ring of rational integers] Z. As is discussed in the introduction to [Mzk8], this kind of result is motivated partly by the an- abelian philosophy of Grothendieck, but perhaps more essentially by the idea that instead of working with set-theoretic objects, such as schemes or log schemes, one should regard categories — which may be thought of as “abstract combinatorial” mathematical objects constituted by some abstract collection of arrows — as the “fundamental, primitive objects” of mathematics discourse. Thus, Grothendieck’s anabelian philosophy may be regarded as a “special case” of this point of view, i.e., the case where the categories in question are Galois categories ⎛ abstract, ⎟ ⎜ ⎜ combinatorial ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ mathematical ⎟ ⎝ ⎠ objects ⎛ ⎞ ⊇ categories ⎛ ⎞ ⎜ ⎟ ⎜ — i.e., abstract ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ collections ⎟ ⎠ ⎝ ⊇ ⎞ ⎟ ⎜ ⎜ Galois ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ categories ⎟ ⎝ ⎠ of arrows — cf. the “absolute anabelian geometry” developed in [Mzk5], [Mzk6], [Mzk7], [Mzk10], [Mzk11], [Mzk12], [Mzk14]. One important drawback of the “anabelian branch” of this category-theoretic approach to mathematics is that although it is very well-suited to capturing essential aspects of the geometry of schemes at nonarchimedean primes, it is ill-suited to capturing the archimedean aspects of the geometry of schemes, and, in particular, those aspects of the global geometry of schemes over number ﬁelds — such as heights — that are of interest in Diophantine geometry. Thus, from this point of view, the extension given in [Mzk9] of the theory of [Mzk8] has the virtue, relative to anabelian geometry, of providing a natural way to incorporate such archimedean and global phenomena as the global degree of an arithmetic line bundle over a number ﬁeld [cf. [Mzk9], Example 5.1] into the above-mentioned category-theoretic approach to mathematics. The approach of [Mzk9], however, has the following fundamental drawback: The categories of [Mzk9] are quite “large” and “complicated” by comparison to Galois categories, in the sense that they include a very diverse collection of arithmetic schemes, by comparison to the ﬁnite étale coverings of a ﬁxed scheme. This makes it relatively easy to reconstruct the original arithmetic log scheme from the category. On the other hand, this relative ease of reconstruction is a reﬂection, in essence, of the fact that the geometry of such categories is really not so diﬀerent from the conventional geometry of arithmetic log schemes. Thus, in other words, one doesn’t gain very much in the way of essentially new geometric phenomena by working with such categories, relative to the conventional geometry of arithmetic log schemes. By contrast, the relatively simple structure of Galois categories [cf. also the categories of [Mzk13]] makes it much more diﬃcult to reconstruct the scheme from the category — indeed, such a reconstruction is only possible in the case of very 6 SHINICHI MOCHIZUKI special “anabelian” schemes — but, on the other hand, this diﬃculty of reconstruc- tion may be regarded as a reﬂection of the fact that there is indeed some interesting new geometry that arises from working with Galois categories that does not exist in the conventional geometry of schemes. Perhaps the most fundamental exam- ple of this phenomenon is the well-known fact that the absolute Galois groups of non-isomorphic ﬁnite ﬁelds are isomorphic. Another less elementary example of this phenomenon is the well-known fact that the Galois category associated to a nonarchimedean mixed-characteristic local ﬁeld [i.e., a ﬁnite extension of the p-adic number ﬁeld] admits self-equivalences [i.e., the associated absolute Galois group admits automorphisms] that do not arise from scheme theory [i.e., from an isomor- phism of ﬁelds — cf., e.g., [NSW], p. 674]. Put another way, the diﬀerence between the “geometry of categories” — i.e., the approach to arithmetic geometry constituted by working with the strictly category- theoretic properties of categories — and the classical approach to arithmetic geom- etry constituted by working with set-theoretic objects equipped with various compli- cated auxiliary structures may be regarded as analogous to the diﬀerence between working with the notion of an abstract group and working with groups of explicit matrices. That is to say, working with strictly group-theoretic properties of abstract groups allows one to contemplate various structures that are common to various distinct groups of explicit matrices, but which are not so evident if one happens to be ignorant of the notion of an “abstract group” and hence obliged to restrict oneself to manipulations involving explicit matrices. This state of aﬀairs prompts the following question: Can one perhaps represent certain special arithmetic log schemes of inter- est by categories whose “level of complexity” is closer to Galois categories [i.e., substantially simpler than the categories of [Mzk9]] — thus allowing one to hope that the geometry of such categories exhibits fundamentally new phenomena that do not appear in the conventional geometry of arith- metic log schemes — on the one hand, but which nevertheless allow one to work naturally with archimedean primes and heights on the other? This sort of question constituted one of the principal motivations for the author to develop the theory discussed in the present paper. The answer to the above question constituted by the theory of present paper is, in a word, the notion of a Frobenioid. From the point of view of the question posed above: Frobenioids provide a single framework [cf. the notion of a “Galois cate- gory”; the role of monoids in log geometry] that allows one to capture the essential aspects of both the Galois and the divisor theory of number ﬁelds, on the one hand, and function ﬁelds, on the other, in such a way that one may continue to work with, for instance, global degrees of arithmetic line bundles on a number ﬁeld, but which also exhibits the new phenome- non [not present in the classical theory of number ﬁelds] of a “Frobenius endomorphism” of the Frobenioid associated to a number ﬁeld. THE GEOMETRY OF FROBENIOIDS I 7 Here, we remark that the base category D is typically a category that is of a level of “simplicity” [cf. the above discussion] that is reminiscent of a Galois category [cf. also the “temperoids” of [Mzk11]; the categories of Riemann surfaces discussed in [Mzk13], §2]. Indeed, in the examples of §6, the base category is [the subcategory of connected objects of] a Galois category. From this point of view, the main ingredients of a Frobenioid — that is to say, roughly speaking, “Galois” [i.e., the base category D], “Frobenius” [i.e., “N ≥1 ”], and “metrics/integral structures” [i.e., the family of monoids Φ] — are reminiscent of the theory of the “ring of p-adic periods” B crys of p-adic Hodge theory. §I3. Frobenius Endomorphisms of a Number Field From a somewhat less conceptual point of view, one of the main motivations for the author in developing the theory of Frobenioids came from the long-term goal of developing a sort of arithmetic Teichmüller theory for number ﬁelds equipped with an elliptic curve, in a fashion that is analogous to the p-adic Teichmüller theory of [Mzk1], [Mzk2]. That is to say, here one wishes to regard number ﬁelds as corresponding to hyperbolic curves over ﬁnite ﬁelds and elliptic curves [over a number ﬁeld] as corresponding to the nilpotent ordinary indigenous bundles [on a hyperbolic curve over a ﬁnite ﬁeld] of [Mzk1], [Mzk2]. In the p-adic Teichmüller theory of [Mzk1], [Mzk2], certain canonical Frobenius liftings play a central role. Thus, since Frobenius liftings are, literally, liftings of the Frobenius morphism in positive characteristic, in order to develop an “arithmetic Teichmüller theory” for number ﬁelds equipped with an elliptic curve, one must ﬁrst have an analogue for number ﬁelds of the Frobenius morphism in positive characteristic scheme theory. If one starts to consider such an analogue from a completely naive point of view, then one must contend with the fact that, if, for instance, n ≥ 2 is a integer, then the morphism p → p n [where p is a prime number] clearly does not extend to a ring homomorphism Z → Z! That is to say, it is diﬃcult to see how to accommodate such a “Frobenius morphism for number ﬁelds” within the framework of scheme theory. On the other hand, if one works with monoids as in the theory of log schemes, then such a morphism “p → p n ” does indeed make sense. Moreover, even if, for instance, one considers roots π of p, the mapping π → π n is Galois-equivariant. Thus, in summary: One important motivation for the author in developing the theory of Frobenioids was the goal of developing a geometric framework — i.e., roughly speaking, a geometry built up solely from “Galois theory” and “monoids” — in which a “Frobenius morphism on number ﬁelds” may be constructed. 8 SHINICHI MOCHIZUKI Once one has constructed such a “Frobenius morphism on number ﬁelds”, the next step to realizing an “arithmetic Teichmüller theory” consists of construct- ing a “canonical Frobenius lifting”. Although the construction of such “canonical Frobenius liftings” lies [well!] beyond the scope of the present paper, we remark that the ideas that lie behind such a construction are motivated by the [scheme- theoretic!] Hodge-Arakelov theory of elliptic curves surveyed in [Mzk3], [Mzk4], a theory in which the theta function on a Tate curve plays a central role. In partic- ular, in order to construct “canonical Frobenius liftings”, it is necessary to extract the essential “abstract, combinatorial content” of the scheme-theoretically formu- lated Hodge-Arakelov theory of [Mzk3], [Mzk4]. In fact, certain aspects of such an “extraction process” are achieved precisely by applying the theory of Frobenioids, as is done in a certain sequel to the present paper and [Mzk15] — namely, [Mzk16]. Here, we pause to observe that to pass from the geometry of schemes to, say, the geometry of Frobenioids amounts to a certain “partial dismantling of scheme theory”, i.e., to “forgetting” a certain portion of scheme theory. As discussed above, one wants to execute such a “partial dismantling of scheme theory” precisely in order to allow the construction of such objects as a “Frobenius morphism on number ﬁelds” which are not possible within the framework of scheme theory. On the other hand, if the dismantling process that one executes is too drastic, then there is a danger of destroying so much of the geometry of scheme theory that one is not left with a geometry that is suﬃciently rich so as to allow the further development of the theory. From this point of view, one of the main themes of the present paper [and [Mzk15]] consists of verifying that: The geometry of Frobenioids retains a substantial portion of the geometry of scheme theory and, in particular, is suﬃciently rich so as to permit the execution of many geometric constructions and arguments familiar from scheme theory. The centerpiece of this veriﬁcation process is the reconstruction of the functor “C → F Φ ”, as discussed in §I1. Another aspect of this veriﬁcation process, which may be seen throughout the theory of the present paper, is the step-by-step translation of various scheme-theoretic terms and constructions that appear in the theory of divisors and line bundles on models of ﬁnite separable extensions of a given function ﬁeld or number ﬁeld into purely category-theoretic language. For instance, one important example of this “step-by-step translation” is the theory of base sections and base-Frobenius pairs developed in §5, which may be thought of as a sort of category-theoretic translation of the notion of the tautological section of a trivial line bundle [cf. Remark 5.6.1]. §I4. Étale-like vs. Frobenius-like Structures Finally, let us return to the “main result” discussed in §I1, i.e., the reconstruc- tion of the functor “C → F Φ ”. One way to think about this result is that it is a statement to the eﬀect that: THE GEOMETRY OF FROBENIOIDS I 9 The structure of a [“permissible”] base category D [e.g., the subcategory of connected objects of a Galois category] is fundamentally combina- torially diﬀerent — indeed, diﬀerent in a category-theoretically distin- guishable fashion — from the structure of the “Frobenius portion” F of a Frobenioid. This phenomenon may be thought of as a sort of fundamental dichotomy between types of combinatorial structures — i.e., between “étale-like” structures which are “indiﬀerent to order” [cf. the ﬁnite groups that as appear as Galois groups in a Galois category] and “Frobenius-like” structures which are “order-conscious” [cf. the monoids “Z ≥0 ”, “N ≥1 ” that constitute the standard Frobenioid F]. One may also think of “étale-like” structures as “descent-compatible” structures, whereas “Frobenius-like” structures are “descent-incompatible”, in the sense that compati- bility with “descent” may be thought of as a sort of violation of the “order” con- stituted by “the object upstairs” in the descent operation and the “the object downstairs”. Relative to the theme of “abstract combinatorialization” discussed in §I2, the point here is that the diﬀerence between “étale-like” and “Frobenius-like” structures is an intrinsic structural diﬀerence, not just a matter of “arbitrar- ily imposed labels motivated by scheme theory” [such as “base category”, “divisor monoid”, “Frobenius degree”, etc.]! For more on this fundamental dichotomy be- tween “étale-like” and “Frobenius-like” categorical structures, we refer to Remark 3.1.3. Acknowledgements: I would like to thank Akio Tamagawa, Makoto Matsumoto, Kazuhiro Fujiwara, and Seidai Yasuda for various useful comments. Also, I would like to thank Yuichiro Taguchi for inviting me to speak at Kyushu University during the Summer of 2003 on a preliminary version of the theory discussed in this paper. 10 SHINICHI MOCHIZUKI Section 0: Notations and Conventions Sets: If E is a partially ordered set, then we shall denote by Order(E) the category whose objects are elements e ∈ E, and whose morphisms e 1 → e 2 [where e 1 , e 2 ∈ E] are the relations e 1 ≤ e 2 . Numbers: We denote by N ≥1 the [discrete] multiplicative monoid of rational integers ≥ 1 and by Primes the set of prime numbers. Thus, one may think of N ≥1 as the free commutative monoid generated by Primes. We shall write: def def R >0 = {a ∈ R | a > 0} ⊆ R ≥0 = {a ∈ R | a ≥ 0} ⊆ R We shall refer to an element Λ ∈ {Z, Q, R} def def def as a monoid type and write Λ >0 = Λ R >0 ⊆ R, Λ ≥0 = Λ R ≥0 ⊆ R, N = Z ≥0 . Also, we shall refer to a monoid isomorphic to [the additive monoid] Λ ≥0 as a Λ- monoprime monoid and to a monoid which is a Λ-monoprime monoid for some Λ as monoprime. If M is a Q-monoprime monoid, then we shall write M ⊗ R ≥0 for the R-monoprime monoid obtained by completing M relative to the topology deﬁned by the ordering on the monoid M . We shall refer to as a number ﬁeld any ﬁnite extension of the ﬁeld of rational numbers. Monoids: Observe that any [not necessarily commutative!] monoid M may be thought of as a special type of category, i.e., the category with precisely one object whose endomorphisms are given by the monoid M . THE GEOMETRY OF FROBENIOIDS I 11 Write Mon for the category of commutative monoids [relative to some universe ﬁxed throughout the discussion]. Let M be an object of Mon; the monoid operation of M will be written additively. We shall denote by M ± ⊆ M the submonoid [which, in fact, forms a group] of invertible elements of M , by M M char = M/M ± def the quotient monoid of M by M ± , which we shall refer to as the characteristic of M , and by M → M gp the natural homomorphism from M to its groupiﬁcation M gp . Thus, M gp is the monoid [which is, in fact, a group] given by the set of equivalence classes of pairs (a, b) ∈ M × M , where two such pairs (a 1 , b 1 ); (a 2 , b 2 ) are considered equivalent if a 1 + b 2 + c = b 1 + a 2 + c, for some c ∈ M , and the monoid operation on this set is the monoid operation induced by the monoid operation of M . We shall say that M is torsion-free if M has no torsion elements; we shall say that M is sharp if M ± = 0; we shall say that M is integral if the natural map M → M gp is injective; we shall say that M is saturated if every a ∈ M gp for which n · a lies in the image of M for some n ∈ N ≥1 lies in the image of M . Denote by M pf the perfection of M , that is to say, the inductive limit of the inductive system I ∗ of monoids n· . . . → M −→M → . . . given by assigning to each element of a ∈ N ≥1 a copy of M , which we denote by I a , and to every two elements a, b ∈ M such that a divides b the morphism def I a = M → I b = M given by multiplication by n = b/a. Thus, the object I 1 of the inductive system I ∗ determines a natural morphism M → M pf which is injective if M is torsion-free, integral, and saturated, hence, in particular, if M is sharp, integral, and saturated. We shall say that M is perfect if multiplication by any element of N ≥1 on M is bijective. Thus, M pf is always perfect; M is perfect if and only if the natural map M → M pf is an isomorphism. Note that M is saturated if and only if M char is. We shall say that M is of characteristic type if the ﬁbers of the natural map M → M char are torsors over M ± . Note that if M is of characteristic type, then M is integral if and only if M char is. If φ : M 1 → M 2 is a morphism of Mon, then we shall say that φ is characteristically injective if φ is injective, and, moreover, the morphism M 1 char → M 2 char induced by φ is injective. 12 SHINICHI MOCHIZUKI Now suppose that M is sharp, integral, and saturated. If a, b ∈ M , then we shall write a ≤ b if ∃ c ∈ M such that a + c = b and a b if ∃n ∈ N ≥1 such that a ≤ n · b. If a subset S ⊆ M satisﬁes the property that there exists a b ∈ M such that a ≤ b for all a ∈ S, then we shall say that S is bounded [by b]. If S ⊆ M is a subset and b ∈ M , then we shall write def Bound S (b) = {a ∈ S | a ≤ b} [i.e., Bound S (b) is the maximal subset of S that is bounded by b]. Observe that if M is R-monoprime, then every bounded subset S ⊆ M possesses a [unique] supremum sup(S) ∈ M [i.e., S is bounded by b if and only if b ≥ sup(S)]. We shall say that 0 = a ∈ M is irreducible if any equation a = b + c in M , where b, c ∈ M , implies that b = 0 or c = 0. We shall say that 0 = a ∈ M is primary if for any M b a, where b = 0, it holds that a b. Denote by Primary(M ) the set of primary elements of M . One veriﬁes immediately that the relation “a b” [where a, b ∈ Primary(M )] determines an equivalence relation on Primary(M ). A -equivalence class of elements of Primary(M ) will be referred to as a prime of M . [Note that this notion of a “prime” diﬀers from the conventional notion of a “prime ideal” of M .] Denote by Prime(M ) the set of primes of M . If p ∈ Prime(M ), then we shall denote by M p ⊆ M the submonoid generated by the elements contained in the subset p ⊆ M . Note that each subset p ⊆ M , where p ∈ Prime(M ), is closed under multiplication by elements of N ≥1 , and that Primary(M pf ) = {a ∈ M pf | ∃n ∈ N ≥1 such that n · a ∈ Primary(M )} Primary(M ) = Primary(M pf ) M ∼ Prime(M ) → Prime(M pf ) [where we regard M as a subset of M pf via the natural inclusion]. Finally, we observe that the relation “≤” on elements of M determines a category Order(M ) [via the partially ordered set structure on M determined by “≤”]. THE GEOMETRY OF FROBENIOIDS I 13 Topological Groups: Let G be a Hausdorﬀ topological group, and H ⊆ G a closed subgroup. Let us write def Z G (H) = {g ∈ G | g · h = h · g, ∀ h ∈ H} for the centralizer of H in G. If Π is a proﬁnite group, then we shall write B(Π) for the category whose objects are ﬁnite sets equipped with a continuous Π-action and whose morphisms are morphisms of Π-sets. Thus, B(Π) is a Galois category, or, in the terminology of [Mzk7], a connected anabelioid. If Z Π (H) = {1} for every open subgroup H ⊆ Π, then we shall say that Π is slim. Categories: Let C be a category. We shall denote the collection of objects (respectively, arrows) of C by: Ob(C) (respectively, Arr(C)) The opposite category to C will be denoted by C opp . A category with precisely one object will be referred to as a one-object category; a category with precisely one morphism [which is necessarily the identity morphism of the unique object of such a category] will be referred to as a one-morphism category. Thus, a one-morphism category is always a one-object category. If A ∈ Ob(C) is an object of C, then we shall denote by C A the category whose objects are morphisms B → A of C and whose morphisms [from an object B 1 → A to an object B 2 → A] are A-morphisms B 1 → B 2 in C and by A C the category whose objects are morphisms A → B of C and whose morphisms (from an object A → B 1 to an object A → B 2 ) are morphisms B 1 → B 2 in C that are compatible with the given arrows A → B 1 , A → B 2 . Thus, we have a natural functor (j A ) ! : C A → C [given by forgetting the structure morphism to A]. Similarly, if f : A → B is a morphism in C, then f deﬁnes a natural functor f ! : C A → C B 14 SHINICHI MOCHIZUKI by mapping an arrow [i.e., an object of C A ] C → A to the object of C B given by the composite C → A → B with f . We shall call an object A ∈ Ob(C) terminal (respectively, pseudo-terminal) if for every object B ∈ Ob(C), there exists a unique arrow (respectively, there exists a [not necessarily unique!] arrow) B → A in C. We shall say that two arrows of a category are co-objective if their domains and codomains coincide. We shall say that an arrow β : B → A of a category C is ﬁberwise-surjective if, for every arrow γ : C → A of C, there exist arrows δ B : D → B, δ C : D → C such that β ◦ δ B = γ ◦ δ C . An arrow of a category which is a ﬁberwise-surjective monomorphism will be referred to as an FSM-morphism. One veriﬁes immediately that every composite of FSM-morphisms is again an FSM-morphism. A category C which satisﬁes the property that every FSM-morphism of C is, in fact, an isomor- phism will be referred to as a category of FSM-type. Let C be a category; A ∈ Ob(C). Write End C (A); Aut C (A) for the monoids of endomorphisms and automorphisms of A in C, respectively. We shall say that an endomorphism α ∈ End C (A) of C is a sub-automorphism if there exists an arrow φ : B → A of C and an automorphism β ∈ Aut C (B) such that φ ◦ β = α ◦ φ; write (Aut C (A) ⊆) Aut sub C (A) ⊆ End C (A) for the subset of End C (A) determined by the sub-automorphisms of A. We shall say that A is Aut-saturated (respectively, Aut sub -saturated; of Aut-type) if Aut C (A) = sub Aut sub C (A) (respectively, Aut C (A) = End C (A); Aut C (A) = End C (A)). If every object of C is Aut-saturated (respectively, Aut sub -saturated; of Aut-type), then we shall say that C is Aut-saturated (respectively, Aut sub -saturated; of Aut-type). We shall say that an arrow A → B of C is an End-equivalence if there exists an arrow B → A in C. We shall refer to a natural transformation between functors all of whose com- ponent morphisms are isomorphisms as an isomorphism between the functors in question. If φ : C 1 → C 2 is a functor between categories C 1 , C 2 , then we shall denote by Aut(φ) — or, when there is no fear of confusion, Aut(C 1 → C 2 ) — the group of automorphisms of the functor φ, and by End(φ) — or, when there is no fear of confusion, End(C 1 → C 2 ) — the monoid of natural transformations from the functor φ to itself. We shall say that φ is rigid if Aut(φ) is trivial. A category C will be called slim if the natural functor C A → C is rigid, for every A ∈ Ob(C). We recall that if Π is a proﬁnite THE GEOMETRY OF FROBENIOIDS I 15 group, then Π is slim if and only if the category B(Π) is slim [cf. [Mzk7], Corollary 1.1.6]. A diagram of functors between categories will be called 1-commutative if the various composite functors in question are isomorphic. When such a diagram “com- mutes in the literal sense” we shall say that it 0-commutes. Note that when a dia- gram in which the various composite functors are all rigid “1-commutes”, it follows from the rigidity hypothesis that any isomorphism between the composite functors in question is necessarily unique. Thus, to state that such a diagram 1-commutes does not result in any “loss of information” by comparison to the datum of a speciﬁc isomorphism between the various composites in question. A category C will be called a skeleton if any two isomorphic objects of C are, in fact, equal. A skeletal subcategory of a category C is a full subcategory S ⊆ C such that S is a skeleton, and, moreover, the inclusion functor S → C is an equivalence of categories. We shall say that a nonempty [i.e., non-initial] object A ∈ Ob(C) is connected if it is not isomorphic to the coproduct of two nonempty objects of C. We shall say that an object A ∈ Ob(C) is mobile if there exists an object B ∈ Ob(C) such that the set Hom C (A, B) has cardinality ≥ 2 [i.e., the diagonal from this set to the prod- uct of this set with itself is not bijective]. We shall say that an object A ∈ Ob(C) is quasi-connected if it is either immobile [i.e., not mobile] or connected. Thus, con- nected objects are always quasi-connected. We shall say that a category C is totally (respectively, almost totally) epimorphic if every morphism in C whose domain is arbitrary (respectively, nonempty) and whose codomain is arbitrary (respectively, connected) is an epimorphism. We shall say that C is of ﬁnitely (respectively, countably) connected type if it is closed under formation of ﬁnite (respectively, countable) coproducts; every object of C is a coproduct of a ﬁnite (respectively, countable) collection of connected objects; and, moreover, all ﬁnite (respectively, countable) coproducts A i in the category satisfy the condition that the natural map Hom C (B, A i ) → Hom C (B, A i ) is bijective, for all connected B ∈ Ob(C). If C is of ﬁnitely or countably connected type, then every nonempty object of C is mobile; in particular, a nonempty object of C is connected if and only if it is quasi-connected. If a mobile object A ∈ Ob(C) satisﬁes the condition that every morphism in C whose domain is nonempty and whose codomain is A is an epimorphism, then A ∼ is connected. [Indeed, C 1 C 2 → A, where C 1 , C 2 are nonempty, implies that the composite map Hom C (A, B) → Hom C (A, B) × Hom C (A, B) → Hom C (C 1 , B) × Hom C (C 2 , B) = Hom C (C 1 ∼ C 2 , B) → Hom C (A, B) is bijective, for all B ∈ Ob(C).] In particular, it follows that if C is a totally epimorphic category, then every object of C is quasi-connected. 16 SHINICHI MOCHIZUKI If C is a category of ﬁnitely or countably connected type, then we shall write C 0 ⊆ C for the full subcategory of connected objects. [Note, however, that in general, objects of C 0 are not necessarily connected — or even quasi-connected — as objects of C 0 !] On the other hand, if, in addition, C is almost totally epimorphic, then C 0 is totally epimorphic [so every object of C is quasi-connected]. If C is a category, then we shall write C ⊥ (respectively, C ) for the category formed from C by taking arbitrary “formal” [possibly empty] ﬁnite (respectively, countable) coproducts of objects in C. That is to say, we deﬁne the “Hom” of C ⊥ (respectively, C ) by the following formula: Hom( def A i , i B j ) = j Hom C (A i , B j ) i j [where the A i , B j are objects of C]. Thus, C ⊥ (respectively, C ) is a category of ﬁnitely (respectively, countably) connected type. Note that objects of C deﬁne connected objects of C ⊥ or C . Moreover, there are natural [up to isomorphism] equivalences of categories ∼ (C ⊥ ) 0 → C; ∼ (C ) 0 → C; ∼ (D 0 ) ⊥ → D; ∼ (E 0 ) → E for D (respectively, E) a category of ﬁnitely connected type (respectively, category of countably connected type). If C is a totally epimorphic category, then C ⊥ (re- spectively, C ) is an almost totally epimorphic category of ﬁnitely (respectively, countably) connected type. In particular, the operations “0”, “⊥” (respectively, “”) deﬁne one-to-one correspondences [up to equivalence] between the totally epimorphic categories and the almost totally epimorphic categories of ﬁnitely (respectively, countably) con- nected type. We observe in passing that if C is a totally epimorphic category, and α ◦ β [where α, β ∈ Arr(C)] is an isomorphism, then α, β are isomorphisms. If C is a [small] category, then we shall write G(C) for the graph associated to C. This graph is the graph with precisely one vertex for each object of C and precisely one edge for each arrow of C [joining the vertices corresponding to the domain and codomain of the arrow]. We shall refer to the full subcategories of C determined by the objects and arrows that compose a connected component of the graph G(C) as a connected component of C. In particular, we shall say that C is connected if G(C) is connected. [Note that by working with respect to some “suﬃciently large” enveloping universe, it makes sense to speak of a category which is not necessarily small at being connected.] THE GEOMETRY OF FROBENIOIDS I 17 Given two arrows f i : A i → B i (where i = 1, 2) in a category C, we shall refer to a commutative diagram ∼ A 1 → A 2 ⏐ ⏐ ⏐ f ⏐ f 1 B 1 2 ∼ → B 2 — where the horizontal arrows are isomorphisms in C — as an abstract equivalence from f 1 to f 2 . If there exists an abstract equivalence from f 1 to f 2 , then we shall say that f 1 , f 2 are abstractly equivalent. If C 1 , C 2 , and D are categories, and Φ 1 : C 1 → D; Φ 2 : C 2 → D are functors, then we deﬁne the “CFP” — i.e., “categorical ﬁber product” — C 1 × D C 2 of C 1 , C 2 over D to be the category whose objects are triples ∼ (A 1 , A 2 , α : Φ 1 (A 1 ) → Φ 2 (A 2 )) where A i ∈ Ob(C i ) (for i = 1, 2); α is an isomorphism of D; and whose morphisms ∼ ∼ (A 1 , A 2 , α : Φ 1 (A 1 ) → Φ 2 (A 2 )) → (B 1 , B 2 , β : Φ 1 (B 1 ) → Φ 2 (B 2 )) are pairs of morphisms γ i : A i → B i [in C i , for i = 1, 2] such that β ◦ Φ 1 (γ 1 ) = Φ 2 (γ 2 )◦α. One veriﬁes easily that if Φ 2 is an equivalence, then the natural projection functor C 1 × D C 2 → C 1 is also an equivalence. Let C be a category; S a collection of arrows in C; φ ∈ Arr(C). Then we shall say that φ is minimal-adjoint to S (respectively, minimal-coadjoint to S; mid-adjoint to S) if every factorization φ = α ◦ β (respectively, φ = β ◦ α; φ = α ◦ β ◦ γ) of φ in C such that β lies in S satisﬁes the property that β is, in fact, an isomorphism. If φ admits a factorization φ = α ◦ β ◦ γ in C, then we shall say that β is subordinate to φ. If φ is not an isomorphism, but, for every factorization φ = α ◦ β in C, it holds that either α or β is an isomorphism, then we shall say that φ is irreducible. We shall refer to an FSM-morphism which is irreducible as an FSMI-morphism. Thus, a category of FSM-type does not contain any FSMI-morphisms. We shall say that a category C is of FSMFF-type [i.e., “FSM-ﬁnitely factorizable type”] if the following two conditions hold: (a) every FSM-morphism of C which is not an isomorphism factors as a composite of ﬁnitely many FSMI-morphisms; (b) for every A ∈ Ob(C), there exists a natural number N such that for every composite φ n ◦ φ n−1 ◦ . . . ◦ φ 2 ◦ φ 1 18 SHINICHI MOCHIZUKI of FSMI-morphisms φ 1 , . . . , φ n such that the domain of φ 1 is equal to A, it holds that n ≤ N . Thus, if C is of FSM-type, then it is of FSMFF-type. Also, we observe that [by condition (b)] no endomorphism of an object of a category of FSMFF-type is an FSMI-morphism. If C is a totally epimorphic category, A ∈ Ob(C), and G ⊆ Aut C (A) is a subgroup, then we shall say that an arrow φ : A → B of C is a categorical quotient of A by G if the following conditions hold: (a) φ ◦ γ = φ, for all γ ∈ G; (b) for every morphism ψ : A → C such that ψ ◦ γ = ψ for all γ ∈ G, there exists a unique morphism ψ : B → C such that ψ = ψ ◦ φ. If φ : A → B is a categorical quotient of A by G, then we shall say that A → B is mono-minimal if the following condition holds: For every factorization φ = φ ◦ ζ, where ζ : A → A is a monomorphism such that there exists a subgroup G ⊆ Aut C (A ), together with an isomorphism ∼ G → G that is compatible, relative to ζ, with the respective actions of G, G on A, A [which implies, by total epimorphicity, that φ : A → B is a categorical quotient of A by G ], it holds that ζ is an isomorphism. Thus, [by total epimorphicity] it follows that an isomorphism is always a mono-minimal categorical quotient of its domain by the trivial group. If C is a category, then we shall say that A ∈ Ob(C) is an anchor if there only exist ﬁnitely many isomorphism classes of objects of A C that arise from irreducible arrows A → B. We shall say that A ∈ Ob(C) is a subanchor if there exists an arrow A → B, where B is an anchor. If C is a totally epimorphic category, then we shall say that A ∈ Ob(C) is an iso-subanchor if there exist a subanchor B ∈ Ob(C), a subgroup G ⊆ Aut C (B), and a morphism B → A [in C] which is a mono-minimal categorical quotient of B by G. THE GEOMETRY OF FROBENIOIDS I 19 Section 1: Deﬁnitions and First Properties In the present §1, we discuss the notion of a Frobenioid, which may be thought of as a category whose internal structure behaves roughly like that of an “elemen- tary Frobenioid”. An “elementary Frobenioid” is, in essence, a sort of semi-direct product of the multiplicative monoid N ≥1 [which is to be thought of as a “Frobenius action”] with a system of monoids [which are roughly of the sort that appear in the theory of log structures] on a “base category” [a category which behaves roughly like a Galois category]. We begin by introducing the fundamental notions of “elementary Frobenioids” and “pre-Frobenioids”. Deﬁnition 1.1. (i) We shall say that M ∈ Ob(Mon) is pre-divisorial if it is integral [cf. §0], saturated [cf. §0], and of characteristic type [cf. §0]. Suppose that M is pre- divisorial. Then we shall say that M is group-like if M char is zero; we shall say that M is divisorial if M is sharp [cf. §0]. [Thus, if M is pre-divisorial, then M char is divisorial.] If α is an endomorphism of a pre-divisorial monoid M ∈ Ob(Mon), then we shall say that α is non-dilating if the endomorphism α char of M char induced by α is the identity endomorphism of M char whenever α char (a) a for all primary [cf. §0] a ∈ M char . (ii) Let D be a category. Then we shall refer to a contravariant functor Φ : D → Mon as a monoid on D if the following conditions are satisﬁed: (a) every morphism of monoids α ∗ : Φ(A) → Φ(B) induced by a morphism α : B → A of D is char- acteristically injective [cf. §0]; (b) if α is an FSM-morphism [cf. §0] of D, then α ∗ : Φ(A) → Φ(B) is an isomorphism of monoids. If, moreover, every monoid Φ(A) [as A ranges over the objects of D] (respectively, some monoid Φ(A) [where A ∈ Ob(D)]) satisﬁes some property of monoids [e.g., is pre-divisorial, sharp, etc.], then we shall say that Φ (respectively, A) satisﬁes this property. Note that if Φ is a monoid on D, then Φ determines monoids “Φ char ”, “Φ gp ”, Φ pf ” on D [i.e., by assigning A → Φ(A) char , A → Φ(A) gp , A → Φ(A) pf ], which we shall refer to, respectively, as the characteristic, groupiﬁcation, and perfection of Φ. If Φ is pre- divisorial, then we shall say that Φ is non-dilating if the endomorphisms of Φ(A), where A ∈ Ob(D), induced by endomorphisms ∈ End D (A) are non-dilating. (iii) Let Φ be a monoid on a category D. Then we shall refer to as the elemen- tary Frobenioid associated to Φ the category F Φ 20 SHINICHI MOCHIZUKI deﬁned as follows: The objects of F Φ are the objects of D. If A, B ∈ Ob(F Φ ), whose respective images in D we denote by A D , B D ∈ Ob(D), then a morphism φ : A → B of F Φ is deﬁned to be a collection of data (φ D , Z φ , n φ ) where φ D : A D → B D is a morphism of D; Z φ ∈ Φ(A D ); n φ ∈ N ≥1 . Here, φ D (respectively, A D ) will be referred to as the projection Base(φ) (respectively, Base(A)) of φ (respectively, A) to D; Z φ as the zero divisor Div(φ) of φ; and n φ as def the Frobenius degree deg Fr (φ) of φ. If C D = Base(C) ∈ Ob(D), then the composite of two morphisms φ = (φ D , Z φ , n φ ) : A → B; ψ = (ψ D , Z ψ , n ψ ) : B → C is given as follows: ψ ◦ φ = (ψ D ◦ φ D , φ ∗D (Z ψ ) + n ψ · Z φ , n ψ · n φ ) : A → C Observe that the assignment Φ → F Φ is functorial with respect to homomorphisms of functors [on D] valued in monoids Φ → Φ ; also, we have a natural projection functor: F Φ → D We shall refer to the D as the base category of F Φ . If M ∈ Ob(Mon), then observe that the elementary Frobenioid F Φ M associated to the functor Φ M on any one- morphism [cf. §0] category that assigns to the unique object of the category the monoid M is itself a one-object [cf. §0] category, whose endomorphism monoid we shall denote by F M and refer to as the elementary Frobenioid associated to M . [Thus, the notation “F ” denotes a category (respectively, monoid) when the subscript “” is a functor (respectively, monoid).] More explicitly, the underlying set of F M is the product M × N ≥1 equipped with the monoid structure is given as follows: if a 1 , a 2 ∈ M , n 1 , n 2 ∈ N ≥1 , def then (a 1 , n 1 ) · (a 2 , n 2 ) = (a 1 + n 1 · a 2 , n 1 · n 2 ). Also, we shall write F = F Z ≥0 and refer to F as the standard Frobenioid. (iv) Let D, Φ, F Φ be as in (iii); C a category. Assume further that Φ is divisorial, and that C, D are connected, totally epimorphic categories [cf. §0]. Then we shall refer to a [covariant] functor C → F Φ as a pre-Frobenioid structure on C. The natural projection functor F Φ → D thus restricts to a natural projection functor C→D on C; similarly, the operations “Base(−)”, “Div(−)”, “deg Fr (−)” on F Φ restrict to operations on C which [by abuse of notation] we shall denote by the same notation. THE GEOMETRY OF FROBENIOIDS I 21 We shall refer to the D as the base category of C. By abuse of notation, we shall often regard Φ as a functor on C [i.e., by composing the original functor Φ with the natural projection functor C → D] and apply similar terminology to objects of C and “Φ as a functor on C” to the terminology applied to objects of D and “Φ as a functor on D” [cf. (ii)]. We shall refer to a category C equipped with a pre-Frobenioid structure C → F Φ as a pre-Frobenioid and to the monoid Φ as the divisor monoid of the pre-Frobenioid. Remark 1.1.1. If φ ◦ ψ is a composite of morphisms φ, ψ of a pre-Frobenioid, then the operations “Base(−)”, “Div(−)”, “deg Fr (−)” behave in the following way under composition: Base(φ ◦ ψ) = Base(φ) ◦ Base(ψ) Div(φ ◦ ψ) = (Base(ψ)) ∗ (Div(φ)) + deg Fr (φ) · Div(ψ) deg Fr (φ ◦ ψ) = deg Fr (φ) · deg Fr (ψ) Indeed, this follows immediately from the deﬁnition of an elementary Frobenioid in Deﬁnition 1.1, (iii). Next, we introduce various terms to describe types of morphisms and objects in a pre-Frobenioid. Deﬁnition 1.2. Let Φ be a divisorial monoid on a connected, totally epimorphic category D; C → F Φ a pre-Frobenioid; φ ∈ Arr(C). Write φ : A → B [where def def A, B ∈ Ob(C)]; A D = Base(A) ∈ Ob(D), B D = Base(B) ∈ Ob(D). Then: (i) We shall say that φ is linear if deg Fr (φ) = 1. We shall say that φ is isometric, or, alternatively, an isometry, if Div(φ) = 0 [cf. Deﬁnition 1.1, (iii)]. If ψ ∈ Arr(C) is co-objective with φ [cf. §0], then we shall say that φ, ψ are metrically equivalent if Div(φ) = Div(ψ). (ii) We shall refer to φ as a base-isomorphism (respectively, base-FSM-morphism) if Base(φ) is an isomorphism (respectively, FSM-morphism [cf. §0]) in D. We shall refer to two objects of C that map to isomorphic objects of D as base-isomorphic. We shall refer to φ as a pull-back morphism if the natural transformation of con- travariant functors on C Hom C (−, A) → Hom C (−, B) × Hom D (−,B D )| C (Hom D (−, A D )| C ) [where “| C ” denotes the restriction of a functor on D to a functor on C via the natural projection functor C → D] induced by φ is an isomorphism. If ψ ∈ Arr(C) is co- objective with φ [cf. §0], then we shall say that φ, ψ are base-equivalent (respectively, Div-equivalent) if Base(φ) = Base(ψ) (respectively, Φ(φ) = Φ(ψ)). If A = B [i.e., φ is an endomorphism], then we shall say that φ is a base-identity (respectively, 22 SHINICHI MOCHIZUKI Div-identity) endomorphism if it is base-equivalent (respectively, Div-equivalent) to the identity endomorphism of A. Write O × (A) ⊆ Aut C (A); O (A) ⊆ End C (A) for the submonoids of base-identity linear endomorphisms. (iii) We shall say that φ is a pre-step [a term motivated by the point of view that the only possibly non-isomorphic portion of such a morphism is the “step” constituted by a non-zero zero divisor] if it is a linear base-isomorphism. If φ is a pre-step, then we shall say that it is a step (respectively, a primary pre-step) if φ is not an isomorphism (respectively, if the zero divisor Div(φ) ∈ Φ(A) of φ is a primary [cf. §0] element of the monoid Φ(A)). We shall say that φ is co-angular [a term that arises from a certain “coincidence of angles” that occurs for co-angular morphisms in the case of Frobenioids that arise in an archimedean context — cf. [Mzk15], Deﬁnition 3.1, (iii)] if, for any factorization φ = α ◦ β ◦ γ in C, where α is linear, β is an isometric pre-step, and either α or γ is a base-isomorphism, it follows that β is an isomorphism. We shall say that φ is LB-invertible [i.e., “line bundle- invertible” — a term motivated by the isomorphism induced by such a morphism between the “image line bundle of the domain” and “the line bundle portion of the codomain” in the case of various Frobenioids that arise from arithmetic geometry] if it is co-angular and isometric. We shall say that φ is a morphism of Frobenius type [a term motivated by the fact that, in the case of Frobenioids that arise from arithmetic geometry, such a morphism corresponds to simply “raising to the n-th tensor power” for some n ∈ N ≥1 ] if φ is an LB-invertible base-isomorphism. We shall say that φ is a prime-Frobenius morphism, or, alternatively, a deg Fr (φ)-Frobenius morphism, if it is a morphism of Frobenius type such that deg Fr (φ) ∈ Primes [cf. §0]. (iv) A Frobenius-ample object of C is deﬁned to be an object C such that for any n ∈ N ≥1 , C admits an endomorphism of Frobenius degree n. A Frobenius-trivial object of C is deﬁned to be an object C such that there exists a homomorphism of monoids ζ : N ≥1 → End C (C) which satisﬁes the following properties: (a) the composite of ζ with the map to N ≥1 given by the Frobenius degree is the identity on N ≥1 ; (b) the endomorphisms in the image of ζ are base-identity endomorphisms of Frobenius type. A Div-Frobenius-trivial object of C is deﬁned to be an object C such that there exists a homomorphism of monoids ζ : N ≥1 → End C (C) which satisﬁes the following properties: (a) the composite of ζ with the map to N ≥1 given by the Frobenius degree is the identity on N ≥1 ; (b) the endomorphisms in the image of ζ are Div-identity endomorphisms of Frobenius type. A universally Div-Frobenius-trivial object of C is deﬁned to be an object C such that for every pull-back morphism C → C of C, it follows that C is a Div-Frobenius-trivial object. A quasi-Frobenius-trivial object of C is deﬁned to be an object C such that for any n ∈ N ≥1 , C admits a base-identity endomorphism [which is not necessarily of Frobenius type!] of Frobenius degree n. A sub-quasi-Frobenius-trivial object of C is deﬁned to be an object C such that there exists a co-angular pre-step D → C in C such that D is quasi-Frobenius trivial. A metrically trivial object of C is deﬁned to be an object C such that for any co-angular pre-step C → D, it holds that THE GEOMETRY OF FROBENIOIDS I 23 D is isomorphic to C. A base-trivial object of C is deﬁned to be an object C such that any object D ∈ Ob(C) such that Base(C) ∼ = Base(D) [in D] is, in fact, sub isomorphic to C. An Aut-ample (respectively, Aut -ample; End-ample) object def of C is deﬁned to be an object C such that, if we write C D = Base(C), then sub the natural map Aut C (C) → Aut D (C D ) (respectively, Aut sub C (C) → Aut D (C D ); End C (C) → End D (C D )) is surjective. A perfect object of C is deﬁned to be an object C such that for every n ∈ N ≥1 , it holds that every B ∈ Ob(C) base-isomorphic to C appears as the codomain of a morphism of Frobenius type of Frobenius degree n, and, moreover, for every pair of morphisms of Frobenius type φ 1 : B 1 → B 1 , φ 2 : B 2 → B 2 of Frobenius degree n, where B 1 , B 2 are base-isomorphic to C, and every pre-step ψ : B 1 → B 2 , there exists a unique pre-step ψ : B 1 → B 2 such that ψ ◦ φ 1 = φ 2 ◦ ψ. A group-like object of C is deﬁned to be an object C such that Φ(C) = 0 [or, equivalently, Φ(C) is group-like — cf. the conventions of Deﬁnition 1.1, (i), (ii), (iv)]. A Frobenius-compact object of C is deﬁned to be an object C such that O × (C) is commutative, O × (C) pf = 0, and every element of Aut C (C) that acts on O × (C) pf via multiplication by an element ∈ Q >0 in fact acts trivially on O × (C) pf . A Frobenius-normalized object of C is deﬁned to be an object C such that if φ ∈ End C (C) is a base-identity endomorphism of Frobenius degree d ∈ N ≥1 , and α ∈ O (C), then α d ◦ φ = φ ◦ α. A unit-trivial object of C is deﬁned to be an object C such that O × (C) = {1}. An isotropic object [a term motivated by the archimedean case — cf. [Mzk15], Deﬁnition 3.1, (iii)] of C is deﬁned to be an object C such that any isometric pre-step C → D in C is, in fact, an isomorphism. We shall write C istr ⊆ C for the full subcategory of isotropic objects and C lin ⊆ C; C bs-iso ⊆ C; C pl-bk ⊆ C for the subcategories determined, respectively, by the linear morphisms, base-isomor- phisms, and pull-back morphisms. We shall say that φ : A → B is an isotropic hull [of A] if φ is an isometric pre-step, B is isotropic, and for every morphism γ : A → C, where C is isotropic, there exists a unique morphism β : B → C such that γ = β ◦φ. A Frobenius-isotropic object of C is deﬁned to be an object C such that there exists a morphism of Frobenius type C → D such that D is isotropic. (v) If every object of C is Frobenius-ample (respectively, Frobenius-trivial; Div- Frobenius-trivial; universally Div-Frobenius-trivial; quasi-Frobenius-trivial; sub- quasi-Frobenius-trivial; metrically trivial; base-trivial; Aut-ample; Aut sub -ample; End-ample; perfect; group-like; Frobenius-compact; Frobenius-normalized; unit- trivial; isotropic; Frobenius-isotropic), then we shall say that the pre-Frobenioid C → F Φ is of Frobenius-ample type (respectively, of Frobenius-trivial type; of Div- Frobenius-trivial type; of universally Div-Frobenius-trivial type; of quasi-Frobenius- trivial type; of sub-quasi-Frobenius-trivial type; of metrically trivial type; of base- trivial type; of Aut-ample type; of Aut sub -ample type; of End-ample type; of perfect type; of group-like type; of Frobenius-compact type; of Frobenius-normalized type; of unit-trivial type; of isotropic type; of Frobenius-isotropic type). 24 SHINICHI MOCHIZUKI Remark 1.2.1. The following implications follow formally from the deﬁnitions: pull-back morphism which is a base-isomorphism ⇐⇒ isomorphism base-trivial =⇒ metrically trivial base-identity =⇒ Div-identity universally Div-Frobenius-trivial =⇒ Div-Frobenius-trivial We are now ready to deﬁne the notion of a “Frobenioid”. Deﬁnition 1.3. Let D, Φ, C → F Φ be as in Deﬁnition 1.2. Then we shall say that the pre-Frobenioid C → F Φ [i.e., C equipped with this functor] is a Frobenioid if the following conditions are satisﬁed: (i) (Surjectivity to the Base Category via Pull-back Morphisms) (a) Every iso- morphism class of D arises as the image via the natural projection functor C → D of an isomorphism class of a Frobenius-trivial object of C. (b) If A, B ∈ Ob(C), def ∼ def A D = Base(A), B D = Base(B), and α : A D → B D is an isomorphism, then there exist pre-steps φ : C → A, ψ : C → B such that α = Base(ψ) ◦ Base(φ) −1 . (c) For every A ∈ Ob(C), the fully faithful [cf. the isomorphism of functors appearing in the deﬁnition of a “pull-back morphism” given in Deﬁnition 1.2, (ii)] functor def pl-bk = (C pl-bk ) A → D A D C A def [where A D = Base(A)] determined by the natural projection functor C → D is an equivalence of categories [cf. §0]. (ii) (Surjectivity to N ≥1 via Morphisms of Frobenius Type) For every A ∈ Ob(C), n ∈ N ≥1 , there exists a morphism of Frobenius type φ : A → B in C of Frobenius degree n; moreover, if ψ : A → C is any other morphism of Frobenius type in C of Frobenius degree n, then there exists a(n) [unique — since C is totally ∼ epimorphic] isomorphism β : B → C such that β ◦ φ = ψ. (iii) (Surjectivity to the Divisor Monoid via Co-angular Morphisms) (a) The co-angular morphisms of C are closed under composition. (b) If A → A is a co- angular pre-step of C, then any morphism A → A is co-angular. (c) Given any co-angular pre-step φ : A → B, there exists a [uniquely determined] bijection of monoids ∼ O (A) → O (B) such that O (A) α → β ∈ O (B) implies β ◦ φ = φ ◦ α; moreover, this bijection depends only [among the bijections induced by the various co-angular pre-steps A → B] on Base(φ). (d) Denote by C coa-pre ⊆ C the subcategory determined by the co-angular pre-steps. Then the natural functors A C coa-pre def = A (C coa-pre ) → Order(Φ(A)); def coa-pre C A = (C coa-pre ) A → Order(Φ(A)) opp THE GEOMETRY OF FROBENIOIDS I 25 [obtained by assigning to an arrow φ : A → B the element Div(φ) ∈ Φ(A) and to ∼ an arrow ψ : B → A the element (ψ ∗ ) −1 (Div(ψ)) ∈ Φ(A) [since ψ ∗ : Φ(A) → Φ(B) is a bijection — cf. the fact that ψ is a base-isomorphism!] are equivalences of categories. (iv) (Factorization of Arbitrary Morphisms) Let φ : A → B be a morphism of C. Then: (a) φ admits a factorization φ = α ◦ β ◦ γ where α is an pull-back morphism, β is a pre-step, and γ is a morphism of Frobenius type; this factorization is unique, up to replacing the triple (α, β, γ) by a triple of the form (α ◦ δ, δ −1 ◦ β ◦ , −1 ◦ γ), where δ, are isomorphisms of C. (b) Every pull-back morphism of C is LB-invertible and linear. (v) (Factorization of Pre-steps) Let φ : A → B be a pre-step of C. Then: (a) φ is a monomorphism. (b) φ admits a factorization φ = α ◦ β where α is an isometric pre-step, and β is a co-angular pre-step; this factorization is unique, up to replacing the pair (α, β) by a pair of the form (α ◦ γ, γ −1 ◦ β), where γ is an isomorphism of C. (c) φ admits a factorization φ = α ◦ β , where α is a co-angular pre-step, and β is an isometric pre-step; this factorization is unique, up to replacing the pair (α , β ) by a pair of the form (α ◦ γ , (γ ) −1 ◦ β ), where γ is an isomorphism of C. (vi) (Faithfulness up to Units) Let φ, ψ : A → B be base-equivalent, metrically equivalent co-angular pre-steps of C. Then there exists a [necessarily unique] α ∈ O × (B) such that φ = α ◦ ψ. (vii) (Isotropic Objects) (a) For every A ∈ Ob(C), there exists a [necessarily unique, up to unique isomorphism] isotropic hull A → B. (b) If A ∈ Ob(C) is isotropic, and A → C is a morphism of C, then C is also isotropic. Remark 1.3.1. Note that it follows from Deﬁnition 1.3, (iii), (b), (c), that if C is a Frobenioid, then the monoid O (A) is commutative, for all A ∈ Ob(C). Proposition 1.4. (Co-angular and LB-invertible Morphisms) Let Φ be a divisorial monoid on a connected, totally epimorphic category D; C → F Φ a pre-Frobenioid; φ : A → B a morphism of C. Then: (i) Suppose that the codomain of any arrow of C whose domain is equal to A is isotropic. Then φ is co-angular. In particular, φ is a morphism of Frobenius type if and only if it is an isometric base-isomorphism. (ii) Suppose that C is a Frobenioid. Then φ is a pull-back morphism if and only if it is an LB-invertible linear morphism [i.e., a co-angular linear isometry]. 26 SHINICHI MOCHIZUKI (iii) Suppose that C is a Frobenioid. Then every LB-invertible pre-step is an isomorphism. (iv) Suppose that C is a Frobenioid. Then a morphism φ of C is co-angular if and only if, in the factorization φ = α ◦ β ◦ γ of Deﬁnition 1.3, (iv), (a), the pre-step β is co-angular. (v) Suppose that C is a Frobenioid. Then a morphism φ of C is LB-invertible if and only if it is of the form α ◦ β, where α is a pull-back morphism, and β is a morphism of Frobenius type. Proof. Assertion (i) follows formally from the deﬁnitions of the terms “isotropic”, “isometric pre-step”, “co-angular”, and “morphism of Frobenius type” [cf. Deﬁni- tion 1.2, (i), (iii), (iv)]. As for assertion (ii), if φ is a pull-back morphism, then it follows from Deﬁnition 1.3, (iv), (b), that φ is an LB-invertible linear morphism. Now suppose that φ is LB-invertible and linear. Then by applying Remark 1.1.1 to the factorization of Deﬁnition 1.3, (iv), (a), the fact that φ is a linear isometry implies that φ may be written in the form α ◦ β, where α is a pull-back morphism, and β is an isometric pre-step. On the other hand, since φ is co-angular, it follows that β is an isomorphism, hence that φ is a pull-back morphism, as desired. As- sertion (iii) follows from either the uniqueness of the factorization of pre-steps of Deﬁnition 1.3, (v), (b), or the essential uniqueness of morphisms of Frobenius type of a given Frobenius degree [cf. Deﬁnition 1.3, (ii)]. Next, we consider assertion (iv). If β is co-angular, then since α, γ are co- angular [cf. assertion (ii); Deﬁnition 1.2, (iii)], it follows from Deﬁnition 1.3, (iii), (a), that φ is co-angular. Conversely, if φ is co-angular, and β = β 1 ◦ β 2 ◦ β 3 , where β 2 is an isometric pre-step, then by applying Remark 1.1.1, together with the fact that D is totally epimorphic [cf. the discussion of §0] to this factorization of β, we conclude that β 1 , β 3 are pre-steps, hence that α ◦ β 1 is linear, and that β 3 ◦ γ is a base-isomorphism; thus, the co-angularity of φ = (α ◦ β 1 ) ◦ β 2 ◦ (β 3 ◦ γ) implies that β 2 is an isomorphism, hence that β is co-angular, as desired. Finally, we consider assertion (v). If φ = α◦β, where α is a pull-back morphism, and β is a morphism of Frobenius type, then [since α, β are LB-invertible — cf. assertion (ii); Deﬁnition 1.2, (iii)] it follows from Remark 1.1.1 that φ is isometric and from Deﬁnition 1.3, (iii), (a), that φ is co-angular, hence LB-invertible. Now suppose that φ is LB-invertible, and that we have a factorization φ = α◦β ◦γ, where α, β, and γ are as in Deﬁnition 1.3, (iv), (a). By assertion (iv), β is co-angular; by Remark 1.1.1, β is isometric. Thus, β is an LB-invertible pre-step, hence [cf. assertion (iii)] an isomorphism, as desired. This completes the proof of assertion (v). Remark 1.4.1. We refer to the Chart of Types of Morphisms in a Frobenioid given at the end of the present paper for a summary of the properties of the base category projections, zero divisors, and Frobenius degrees satisﬁed by various types of morphisms in a Frobenioid. THE GEOMETRY OF FROBENIOIDS I 27 Proposition 1.5. (Elementary Frobenioids are Frobenioids) Let Φ be a pre-divisorial monoid on a connected, totally epimorphic category D. Then: (i) F Φ , equipped with the natural functor F Φ → F Φ char , is a Frobenioid of Aut- ample, Aut sub -ample, End-ample, base-trivial, Frobenius-trivial, Frobenius- normalized, and isotropic type. (ii) There is a natural, functorial isomorphism ∼ O (A) → Φ(A) ∼ [so O × (A) → Φ(A) ± ] for objects A ∈ Ob(F Φ ). (iii) If all of the monoids in the image of Φ are perfect (respectively, group- like), then F Φ is of perfect (respectively, group-like) type. Proof. Since D is a connected, totally epimorphic category, the fact that F Φ is as well follows immediately from the deﬁnition of the morphisms of F Φ in Deﬁni- tion 1.1, (iii); the fact that a pre-divisorial monoid is integral [cf. Deﬁnition 1.1, (i)]; and the injectivity condition of Deﬁnition 1.1, (ii), (a). Thus, F Φ is a pre- Frobenioid. It is immediate from the deﬁnitions that assertion (ii) holds, and that all objects of F Φ are Aut-ample, Aut sub -ample, End-ample, base-trivial, Frobenius- trivial, Frobenius-normalized, and isotropic. Also, one veriﬁes immediately [cf. the deﬁnition of the category F Φ in Deﬁnition 1.1, (iii)] that a morphism of F Φ is a pull-back morphism if and only if it is a linear isometry. The fact that F Φ satisﬁes the conditions of Deﬁnition 1.3 now follows immediately from the deﬁnition of the category F Φ in Deﬁnition 1.1, (iii), together with assertion (ii) and the “explicit description” of co-angular morphisms and morphisms of Frobenius type in Propo- sition 1.4, (i) [which is applicable to all morphisms of F Φ since F Φ is of isotropic type]. This completes the proof of assertion (i). Assertion (iii) is immediate from the deﬁnitions and assertion (i). One important technique for constructing new Frobenioids is given by the fol- lowing result. Proposition 1.6. (Categorical Fiber Products) Let Φ be a divisorial monoid on a connected, totally epimorphic category D; C → F Φ a Frobenioid. Let D be a connected, totally epimorphic category; D → D a functor that maps FSM-morphisms to FSM-morphisms. Denote by Φ : D → Mon the divisorial monoid obtained by restricting Φ to D . Then: (i) There is a natural equivalence of categories ∼ F Φ → F Φ × D D [where the latter category is the categorical ﬁber product of §0]. 28 SHINICHI MOCHIZUKI (ii) The categorical ﬁber product [cf. §0] def C = C × D D equipped with the functor C → F Φ [obtained by applying “(−)× D D ” to the functor C → F Φ ] is a Frobenioid. (iii) A morphism of C is a(n) isometry (respectively, morphism of a given Frobenius degree; co-angular morphism; LB-invertible morphism; pull- back morphism) if and only if its projection to C is. (iv) A base-isomorphism of C is a morphism of Frobenius type (respec- tively, pre-step; step) if and only if its projection to C is. Moreover, the projection ∼ functor C → C determines a bijection of monoids O (A ) → O (A), for every A ∈ Ob(C ) that projects to A ∈ Ob(C). (v) A object of C is Frobenius-trivial (respectively, quasi-Frobenius-trivial; sub-quasi-Frobenius-trivial; metrically trivial; base-trivial; perfect; group- like; unit-trivial; Frobenius-normalized; isotropic; Frobenius-isotropic) if and only if it projects to such an object of C. (vi) A object of C is Aut-ample (respectively, Aut sub -ample; End-ample) if it projects to such an object of C. Proof. Assertion (i) follows formally from the deﬁnitions. Next, observe that the fact that D is a totally epimorphic category implies immediately that C is as well; similarly, [in light of the various properties of the natural projection functor C → D assumed in Deﬁnition 1.3, (i), (a), (b), (c)] the fact that D is connected implies immediately that C is also connected. Thus, C [equipped with the functor C → F Φ obtained by applying “(−) × D D ” to the functor C → F Φ ] is a pre- Frobenioid. Now assertion (vi) follows immediately from the deﬁnitions; one checks immediately that the equivalences of assertions (iii), (iv), (v) hold. In light of these equivalences, the conditions of Deﬁnition 1.3 follow via a routine veriﬁcation. Thus, C is a Frobenioid. This completes the proof of assertion (ii). Proposition 1.7. (Composites of Morphisms) Let Φ be a divisorial monoid on a connected, totally epimorphic category D; C → F Φ a Frobenioid. Then: (i) The following classes of morphisms are closed under composition: isome- tries, base-isomorphisms, base-FSM-morphisms, pull-back morphisms, linear morphisms, pre-steps, co-angular morphisms, LB-invertible mor- phisms, morphisms of Frobenius type. (ii) A morphism of C is a pull-back morphism if and only if it is minimal- adjoint to the base-isomorphisms of C. A morphism of C is a base-isomorphism if and only if it is minimal-coadjoint to the pull-back morphisms of C; alter- natively, a morphism of C is a base-isomorphism if and only if it is may be written THE GEOMETRY OF FROBENIOIDS I 29 as a composite α ◦ β, where α is a pre-step, and β is a morphism of Frobenius type. (iii) A morphism of C is of Frobenius type if and only if it is minimal- coadjoint to the linear morphisms of C. A morphism of C is linear if and only if it is minimal-adjoint to the morphisms of Frobenius type of C; alternatively, a morphism of C is linear if and only if it is may be written as a composite α ◦ β, where α is a pull-back morphism, and β is a pre-step. (iv) A pre-step of C is co-angular if and only if it is mid-adjoint [cf. §0] to the isometric pre-steps. (v) If a composite morphism φ = α◦β of C is a(n) isomorphism (respectively, base-isomorphism; linear morphism; pre-step; isometry; co-angular pre- step; co-angular linear morphism; pull-back morphism), then so are α, β. If, moreover, the domain of φ is isotropic, then a similar statement holds for morphisms of Frobenius type. Proof. Assertion (i) follows immediately from the deﬁnitions for isometries, base- isomorphisms, base-FSM-morphisms, pull-back morphisms, linear morphisms, and pre-steps; from Deﬁnition 1.3, (iii), (a), for co-angular morphisms, hence also for LB-invertible morphisms and morphisms of Frobenius type. Next, the suﬃciency of the various conditions given in assertions (ii), (iii) follows immediately from [deﬁni- tions and] the [existence of the] factorization of Deﬁnition 1.3, (iv), (a). Moreover, in light of the existence of this factorization, the necessity of the various conditions given in assertions (ii), (iii) follows immediately for pull-back morphisms and mor- phisms of Frobenius type from the essential uniqueness of this factorization [and the total epimorphicity of C]; for base-isomorphisms from the total epimorphicity of D; and for linear morphisms from the well-known structure of the multiplicative monoid N ≥1 and the essential uniqueness of morphisms of Frobenius type of a given Frobenius degree [cf. Deﬁnition 1.3, (ii)]. In light of Remark 1.1.1, assertion (v) follows for isomorphisms (respectively, base-isomorphisms; linear morphisms; pre-steps; isometries) immediately from the fact that C is totally epimorphic (respectively, from the fact that D is totally epi- morphic; from the well-known structure of the multiplicative monoid N ≥1 ; from assertion (v) for base-isomorphisms and linear morphisms; from the fact that the monoid Φ on D is sharp [cf. Deﬁnition 1.1, (i)], together with the characteristic in- jectivity assumption of Deﬁnition 1.1, (ii), (a)). Now assertion (iv) follows formally from [the deﬁnitions and] assertion (v) for pre-steps [cf. the argument applied in the proof of Proposition 1.4, (iv)!]; assertion (v) for co-angular pre-steps follows from assertion (v) for pre-steps and assertion (iv). To prove assertion (v) for co- angular linear morphisms, suppose that φ is co-angular and linear. Then observe that by assertion (v) for linear morphisms, α, β are linear. Thus, by applying the factorization for linear morphisms of assertion (iii), together with the factorization of Deﬁnition 1.3, (v), (c) [cf. also Proposition 1.4, (ii); assertion (i) for co-angular linear morphisms], we may write α = α 1 ◦ α 2 , β = β 1 ◦ β 2 , α 2 ◦ β 1 = γ 1 ◦ γ 2 , where α 1 , β 1 , γ 1 are co-angular linear morphisms, and α 2 , β 2 , γ 2 are isometric pre-steps. 30 SHINICHI MOCHIZUKI Thus, φ = (α 1 ◦ γ 1 ) ◦ (γ 2 ◦ β 2 ), which [by the co-angularity of φ] implies that γ 2 ◦ β 2 is an isomorphism, hence [by assertion (v) for isomorphisms] that β 2 , γ 2 are isomor- phisms. Thus, by the co-angularity of α 2 ◦ β 1 = γ 1 ◦ γ 2 , we conclude that α 2 is an isomorphism. In particular, it follows that α, β are co-angular linear morphisms, as desired. Now assertion (v) for pull-back morphisms follows from assertion (v) for co-angular linear isometries [cf. also Proposition 1.4, (ii)]. Finally, assertion (v) for morphisms of Frobenius type in C istr [cf. Deﬁnition 1.3, (vii), (b)] follows from assertion (v) for isometric base-isomorphisms, since morphisms of C istr are always co-angular [cf. Proposition 1.4, (i)]. This completes the proof of assertion (v). Proposition 1.8. (Pre-steps) Let Φ be a divisorial monoid on a connected, totally epimorphic category D; C → F Φ a Frobenioid. Then: (i) If the natural projection functor C → D is full, then every pre-step of C is a linear End-equivalence. If D is of Aut-type [cf. §0], then every linear End-equivalence of C is a pre-step. (ii) Suppose further that C is of metrically trivial and Aut-ample type. Then a morphism of C is a co-angular pre-step if and only if it is abstractly equivalent [cf. §0] to a base-identity pre-step endomorphism of C. (iii) An object A ∈ Ob(C) is non-group-like if and only if there exists a co-angular step A → B; alternatively, an object A ∈ Ob(C) is non-group-like if and only if there exists a co-angular step B → A. Also, if A, B ∈ Ob(C) are base-isomorphic objects, then A is group-like if and only if B is. Proof. First, we consider assertion (i). If φ ∈ Arr(C) is a pre-step, and the projection functor C → D is full, then the fact that it is a linear End-equivalence follows formally from the deﬁnition of a “pre-step” [cf. Deﬁnition 1.2, (iii)]; the fullness assumption on C → D. On the other hand, if φ ∈ Arr(C) is a linear End-equivalence, and D is of Aut-type, then it follows formally that φ is a base- isomorphism, hence a pre-step, as desired. This completes the proof of assertion (i). Next, we consider assertion (ii). If φ ∈ Arr(C) is a co-angular pre-step, then it follows formally from the assumption that C is of metrically trivial and Aut-ample type that φ is abstractly equivalent to a base-identity pre-step endomorphism of C. On the other hand, if φ ∈ Arr(C) is abstractly equivalent to a base-identity pre-step endomorphism of C [hence co-angular, by Deﬁnition 1.3, (iii), (b)], then it follows formally that φ is a co-angular linear base-isomorphism, hence that φ is a co-angular pre-step, as desired. This completes the proof of assertion (ii). Finally, we observe that the various equivalences of assertion (iii) follow formally from the deﬁnitions and the equivalences of categories of Deﬁnition 1.3, (iii), (d). Proposition 1.9. (Isotropic Objects and Isometries) Let Φ be a divisorial monoid on a connected, totally epimorphic category D; C → F Φ a Frobenioid. THE GEOMETRY OF FROBENIOIDS I 31 Write C imtr-pre ⊆ C for the subcategory determined by the isometric pre-steps and imtr-pre def = (C imtr-pre ) A C A for A ∈ Ob(C). Then: (i) Any base-isomorphism φ : A → B of C admits a factorization φ = α ◦ β where α is an isometric pre-step, and β is a co-angular base-isomorphism; this factorization is unique, up to replacing the pair (α, β) by a pair of the form (α ◦ γ, γ −1 ◦ β), where γ is an isomorphism of C. Here, φ is isometric if and only if β is a morphism of Frobenius type; φ is co-angular if and only if α is an isomorphism; φ is a pull-back morphism if and only if φ is an isomorphism. (ii) Any base-isomorphism φ : A → B of C induces a functor [well-deﬁned up to isomorphism] imtr-pre imtr-pre φ ∗ : C A → C B that maps an isometric pre-step C → A to the isometric pre-step D → B appearing in the factorization C → D → B of (i) applied to the composite of the given pre-step C → A with φ : A → B. Moreover, if φ is a co-angular pre-step, then φ ∗ is an equivalence of categories. If u ∈ O × (A), then we shall denote by u imtr-pre the imtr-pre induced by u and isomorphism class of the self-equivalence of the category C A by O × (A) imtr-pre ⊆ O × (A) the subgroup of v ∈ O × (A) for which v imtr-pre is the identity. (iii) Any pull-back morphism φ : A → B of C induces a functor [well- deﬁned up to isomorphism] imtr-pre imtr-pre φ ∗ : C B → C A that maps an isometric pre-step δ : D → B to the unique [up to isomorphism] isometric pre-step γ : C → A that ﬁts into a commutative diagram γ C −→ ⏐ ⏐ ψ D δ −→ A ⏐ ⏐ φ B where ψ is the pull-back morphism that arises by applying the equivalence of cat- egories of Deﬁnition 1.3, (i), (c), to the arrow Base(δ) −1 ◦ Base(φ), and γ is the morphism that arises from the isomorphism of functors appearing in the deﬁnition of a “pull-back morphism” [cf. Deﬁnition 1.2, (ii)]. (iv) Let φ : A → B be a co-angular linear morphism [e.g., a pull-back morphism — cf. Proposition 1.4, (ii)]. Then A is isotropic if and only if B is. 32 SHINICHI MOCHIZUKI (v) C istr [equipped with the restriction to C of the given functor C → F Φ ] is a Frobenioid. Moreover, the functor C → C istr that assigns to an object A ∈ Ob(C) with isotropic hull A → A istr the object A istr and to a morphism of objects A → B with isotropic hulls A → A istr , B → B istr the induced [i.e., by the deﬁnition of an “isotropic hull”!] morphism A istr → B istr forms a left adjoint to the inclusion functor C istr → C, through which the functor C → F Φ factors. We shall refer to this functor as the isotropi- ﬁcation functor. The restriction of the isotropiﬁcation functor to C istr is iso- morphic to the identity functor. Finally, the isotropiﬁcation functor preserves morphisms of Frobenius type, Frobenius degrees, pre-steps, pull-back morphisms, base-isomorphisms, base-FSM-morphisms, base-identity en- domorphisms, Div-identity endomorphisms, isometries, co-angular mor- phisms, and LB-invertible morphisms; moreover, all of these properties are compatible with the inclusion functor C istr → C [in the sense that an arrow of C istr satisﬁes one of these properties with respect to C istr if and only if it does with respect to C]. (vi) A morphism of C is an isotropic hull if and only if its codomain is isotropic, and, moreover, it is minimal-coadjoint to the morphisms with isotropic domain. (vii) A morphism A → B of C is an isometric pre-step if and only if the com- posite of this morphism A → B with an isotropic hull B → C yields an isotropic hull A → C. Proof. Since pull-backs which are base-isomorphisms are easily veriﬁed to be isomorphisms [cf. Remark 1.2.1], assertion (i) follows immediately from the (es- sentially) unique factorization of Deﬁnition 1.3, (iv), (a); the (essentially) unique factorization of pre-steps of Deﬁnition 1.3, (v), (b); the fact that co-angular mor- phisms are closed under composition [cf. Proposition 1.7, (i)]; the deﬁnition of “co-angular” [cf. Deﬁnition 1.2, (iii)]; the fact that C is totally epimorphic; the essential uniqueness of morphisms of Frobenius type of a given Frobenius degree [cf. Deﬁnition 1.3, (ii)]; and Remark 1.1.1. Next, we consider assertion (ii). The existence of the functor φ ∗ follows for- mally from the existence of the (essentially) unique factorization of assertion (i). Now suppose that φ is a co-angular pre-step. Then for any isometric pre-step β : D → B, there exists a co-angular pre-step ψ : C → D such that (Φ(β ◦ ψ)) −1 (Div(ψ))) = (Φ(φ)) −1 (Div(φ)) [cf. the second equivalence of categories of Deﬁnition 1.3, (iii), (d)]. Thus, by applying the factorization of Deﬁnition 1.3, (v), (c), it follows that we may write β ◦ ψ = φ ◦ α , where α : D → A is an isometric pre-step, and φ : A → B is a THE GEOMETRY OF FROBENIOIDS I 33 co-angular pre-step. On the other hand, since Div(β ◦ ψ) = Div(φ ◦ α ), and β, α are isometric, it follows that (Φ(φ)) −1 (Div(φ)) = (Φ(φ )) −1 (Div(φ )) — hence [by the second equivalence of categories of Deﬁnition 1.3, (iii), (d)] that ∼ there exists an isomorphism γ : A → A such that φ ◦ γ = φ . Thus, if we take def α = γ ◦α , then β ◦ψ = φ◦α — that is to say, φ ∗ is essentially surjective. Moreover, [by possibly replacing φ by ψ] this argument [i.e., the construction, given β, φ, of α, ψ such that β ◦ ψ = φ ◦ α] also implies that φ ∗ is full. Finally, since every pre-step is a monomorphism [cf. Deﬁnition 1.3, (v), (a)], it follows immediately that φ ∗ is faithful. This completes the proof of assertion (ii). Assertion (iii) follows formally from the deﬁnitions, together with the fact that pull-back morphisms are linear isometries [cf. Proposition 1.4, (ii)], which implies [cf. Remark 1.1.1] that γ is an isometric pre-step. Next, we consider assertion (iv). Let φ : A → B be a co-angular linear mor- phism. If A is isotropic, then so is B, by Deﬁnition 1.3, (vii), (b). Now suppose that B is isotropic. Thus, by the deﬁnition of an isotropic hull, it follows from the existence of isotropic hulls [cf. Deﬁnition 1.3, (vii), (a)] that there exists a factor- ization φ = β ◦ α, where α : A → A is an isotropic hull [hence an isometric pre-step — cf. Deﬁnition 1.2, (iv)], and β : A → B is linear [cf. Remark 1.1.1]. Thus, by the deﬁnition of “co-angular” [cf. Deﬁnition 1.2, (iii)], we conclude that α is an isomorphism, as desired. This completes the proof of assertion (iv). Next, we consider assertion (v). By applying the deﬁnition of an isotropic hull [cf. Deﬁnition 1.2, (iv)], it follows immediately [from the fact that C is connected and totally epimorphic] that C istr is connected and totally epimorphic. Thus, C istr is a pre-Frobenioid. It is immediate from the deﬁnition of an isotropic hull that the isotropiﬁcation functor is left adjoint to the inclusion functor C istr → C; that the functor C → F Φ factors through the isotropiﬁcation functor [cf. Remark 1.1.1]; that the restriction of the isotropiﬁcation functor to C istr is isomorphic to the identity functor; and [cf. Remark 1.1.1] that the isotropiﬁcation functor preserves Frobe- nius degrees, pre-steps, base-isomorphisms, base-FSM-morphisms, base-identity en- domorphisms, Div-identity endomorphisms, isometries, and co-angular morphisms [cf. Proposition 1.4, (i)], hence also LB-invertible morphisms and morphisms of Frobenius type in a fashion that is compatible [cf. the statement of assertion (v)] with the inclusion C istr → C. Since pull-back morphisms are co-angular linear isometries [cf. Proposition 1.4, (ii)], it follows immediately [in light of what we have shown so far] from Proposition 1.4, (ii), that the isotropiﬁcation functor maps pull-back morphisms to morphisms which are pull-back morphisms relative to C, hence a fortiori, pull-back morphisms relative to C istr . Finally, in light of Proposi- tion 1.4, (i); assertion (iv) [cf. also Deﬁnition 1.3, (vii), (b)], it follows immediately [from the fact that C is a Frobenioid!] that the pre-Frobenioid C istr satisﬁes the various conditions of Deﬁnition 1.3, hence that C istr is a Frobenioid, as desired. This completes the proof of assertion (v). Finally, we observe that the necessity and suﬃciency of the condition of asser- tion (vi) follow immediately from the deﬁnition of an isotropic hull [cf. Deﬁnition 34 SHINICHI MOCHIZUKI 1.2, (iv)], the existence of isotropic hulls [cf. Deﬁnition 1.3, (vii), (a)] and the total epimorphicity of C; the necessity and suﬃciency of the condition of assertion (vii) follow immediately from the existence of isotropic hulls [cf. Deﬁnition 1.3, (vii), (a)], the fact that isometric pre-steps between isotropic objects are isomorphisms [cf. Deﬁnition 1.3, (vii), (b); Proposition 1.4, (i), (iii)], and the following observa- tion [which follows immediately from Proposition 1.7, (i), (v)]: Given morphisms α, β, γ of C such that γ = α ◦ β, if any two of the three morphisms α, β, γ is an isometric pre-step, then the same is true of the remaining morphism. Proposition 1.10. (Morphisms of Frobenius Type) Let Φ be a divisorial monoid on a connected, totally epimorphic category D; C → F Φ a Frobenioid. Then: (i) Let φ : A → B be an arbitrary morphism of C. Suppose that α : A → A , β : B → B are morphisms of Frobenius type, of Frobenius degree d ∈ N ≥1 . Then there exists a unique morphism φ : A → B such that the following diagram commutes: φ A −→ B ⏐ ⏐ ⏐ β ⏐ α A φ −→ B In this situation, deg Fr (φ) = deg Fr (φ ); Div(φ ) = d · α ∗ (Div(φ)) [where we write ∼ α ∗ : Φ(A) → Φ(A ) for the bijection induced by applying the functor Φ to the base- isomorphism α]. Finally, if φ is a morphism of Frobenius type (respectively, pre-step; pull-back morphism; co-angular morphism; base-isomorphism; isometry; LB-invertible morphism), then the same is true of φ . (ii) Any composite morphism β ◦ α of C, where α is a pre-step, and β is of Frobenius type, may be written as a composite α ◦ β = β ◦ α where α is a pre-step, and β is of Frobenius type such that: deg Fr (β) = deg Fr (β ); Div(α ) = deg Fr (β) · β ∗ (Div(α)) [where we write β ∗ for the bijection induced by applying the functor Φ to the base- isomorphism β ]. (iii) Suppose that C is of perfect type. Then the monoids in the image of Φ are perfect. If, moreover, C is of isotropic and Frobenius-normalized type, then the monoids O (A) and O × (A) are perfect. (iv) A morphism of Frobenius type with isotropic domain is a prime-Frobe- nius morphism if and only if it is irreducible [cf. §0]. In particular, if A ∈ Ob(C) is isotropic, then there exist inﬁnitely many isomorphism classes of objects of A C that arise from irreducible arrows with domain A. THE GEOMETRY OF FROBENIOIDS I 35 (v) A morphism of C is a morphism of Frobenius type if and only if it is a composite of prime-Frobenius morphisms. (vi) The Frobenioid C istr is of sub-quasi-Frobenius-trivial type. Moreover, every group-like object A ∈ Ob(C istr ) is Frobenius-trivial. Proof. First, we consider assertion (i). Observe that uniqueness follows from the fact that C is totally epimorphic. Now it suﬃces to prove the existence of φ as desired, ﬁrst in the case where φ is a morphism of Frobenius type, then in the case where φ is a pre-step, and ﬁnally in the case where φ is a pull-back morphism [cf. the factorization of Deﬁnition 1.3, (iv), (a)]. In the ﬁrst case, since morphisms of Frobenius type are closed under composition, with multiplying Frobenius degrees [cf. Proposition 1.7, (i); Remark 1.1.1], the existence of a morphism of Frobenius type φ as desired follows immediately from the existence and (essential) uniqueness of morphisms of Frobenius type of a given Frobenius degree [cf. Deﬁnition 1.3, (ii)]. In the case where φ is a pre-step, the existence of a pre-step φ [which, moreover, is co-angular if φ is] as desired follows immediately from the factorization of Deﬁnition 1.3, (iv), (a) [cf. also Proposition 1.4, (iv)], together with the (essential) unique- ness of morphisms of Frobenius type of a given Frobenius degree [cf. Deﬁnition 1.3, (ii)], and the fact that co-angular morphisms are closed under composition [cf. Proposition 1.7, (i)]. In a similar vein, since pull-back morphisms are LB-invertible [cf. Proposition 1.4, (ii)], and LB-invertible morphisms are closed under compo- sition [cf. Proposition 1.7, (i)], the existence of a pull-back morphism φ in the case where φ is a pull-back morphism follows immediately from the factorization of Proposition 1.4, (v), together with the (essential) uniqueness of morphisms of Frobenius type of a given Frobenius degree [cf. Deﬁnition 1.3, (ii)]. The portion of assertion (i) concerning “deg Fr (−)”, “Div(−)” then follows immediately from Remark 1.1.1. Finally, in light of what we have done so far, the fact that “if φ is a(n) co-angular morphism (respectively, base-isomorphism; isometry; LB-invertible morphism), then the same is true of φ ” follows immediately from the deﬁnitions; Remark 1.1.1; the factorization of co-angular morphisms given in Proposition 1.4, (iv); and the fact that co-angular morphisms are closed under composition [cf. Proposition 1.7, (i)]. This completes the proof of assertion (i). Now [in light of the existence of morphisms of Frobenius type of a given Frobenius degree — cf. Deﬁnition 1.3, (ii)] assertion (ii) follows formally from assertion (i). Next, we consider assertion (iii). In light of the existence of morphisms of Frobenius type of a given Frobenius degree [cf. Deﬁnition 1.3, (ii)] and the equiva- lences of categories of Deﬁnition 1.3, (iii), (d), the fact that Φ(A) is perfect follows immediately [cf. Remark 1.1.1] from the fact that A is perfect [cf. Deﬁnition 1.2, (iv)]. Now suppose further that C is of isotropic [so all morphisms of C are co-angular — cf. Proposition 1.4, (i)] and Frobenius-normalized type. Then by the existence of Frobenius-trivial objects [cf. Deﬁnition 1.3, (i), (a), (b); the isomorphism of Def- inition 1.3, (iii), (c)], we may assume that A is Frobenius-trivial. Now the fact that the monoids O (A) and O × (A) are perfect follows immediately from the fact that A is perfect [cf. Deﬁnition 1.2, (iv), applied to the base-identity endomorphisms of Frobenius type of the Frobenius-trivial object A] and Frobenius-normalized. This completes the proof of assertion (iii). 36 SHINICHI MOCHIZUKI Next, we observe that assertion (iv) follows immediately from Proposition 1.7, (v), and the well-known structure of the multiplicative monoid N ≥1 [cf. also Def- inition 1.3, (ii)], and that assertion (v) follows immediately from Proposition 1.7, (i); Deﬁnition 1.3, (ii). Finally, we consider assertion (vi). Let A ∈ Ob(C istr ). Then by Deﬁnition 1.3, (i), (a), (b) [applied to the Frobenioid C istr — cf. Proposition 1.9, (v)], there exist co-angular [cf. Proposition 1.4, (i)] pre-steps α : B → A, γ : B → C, where C is Frobenius-trivial. Thus, for d ∈ N ≥1 , there exists a base-identity endomorphism of Frobenius type φ C ∈ End C (C) such that deg Fr (φ C ) = d; by assertion (ii) [cf. also Proposition 1.4, (i)], we may write φ C ◦ γ = γ ◦ ψ, where ψ : B → B is a morphism of Frobenius type, and γ : B → C is a co-angular pre-step. Moreover, the portion of assertion (ii) concerning the relationship between Div(γ), Div(γ ) implies, in light of the second equivalence of categories of Deﬁnition 1.3, (iii), (d), that γ factors through γ, i.e., there exists a co-angular pre-step β : B → B such that def γ ◦ β = γ . Thus, if we set φ B = β ◦ ψ ∈ End C (B), then γ ◦ φ B = φ C ◦ γ. Moreover, since φ C is a base-identity endomorphism of Frobenius degree d, and γ is a pre- step, it follows [cf. Remark 1.1.1] that φ B is also a base-identity endomorphism of Frobenius degree d. Thus, we conclude that B is quasi-Frobenius-trivial, hence that A is sub-quasi-Frobenius-trivial, as desired. If, moreover, A is group-like, then [since C istr is a Frobenioid — cf. Proposition 1.9, (v)] it follows from Deﬁnition 1.3, (i), (a), (b), that there exist [co-angular — cf. Proposition 1.4, (i)] pre-steps A → A, A → A , where A is Frobenius-trivial. But by Proposition 1.4, (iii), these pre- steps are isomorphisms, so A is Frobenius-trivial, as desired. This completes the proof of assertion (vi). Proposition 1.11. (Pull-back and Linear Morphisms) Let Φ be a divi- sorial monoid on a connected, totally epimorphic category D; C → F Φ a Frobe- nioid. Then: (i) Suppose further that C is of Aut-ample and base-trivial type. Then the natural projection functor C pl-bk → D is full. (ii) Suppose further that C is of unit-trivial type. Then the natural projection functor C pl-bk → D is faithful. (iii) Let φ : B → A be a pull-back morphism that projects to a morphism def φ D = Base(φ) : B D → A D of D . Then given any α ∈ End C (A), β D ∈ End D (B D ) such that Base(α) ◦ φ D = φ D ◦ β D , there exists a unique β ∈ End C (B) such that Base(β) = β D , α ◦ φ = β ◦ φ. (iv) Every co-angular linear morphism φ : B → A determines an injection of monoids O (A) → O (B) which is uniquely determined by the condition that O (A) implies α ◦ φ = φ ◦ β. α → β ∈ O (B) THE GEOMETRY OF FROBENIOIDS I 37 (v) The equivalences of categories of Deﬁnition 1.3, (iii), (d), are “functorial” in the following sense: If φ : A → B is an arbitrary morphism of C lin , α : C → A and β : D → B (respectively, α : A → C and β : B → D) are co-angular pre-steps such that (α ∗ ) −1 (Div(α)) = φ ∗ {(β ∗ ) −1 (Div(β))} (respectively, Div(α) = φ ∗ (Div(β))), then there exists a unique morphism ψ : C → D in C lin such that β ◦ ψ = φ ◦ α (respectively, ψ ◦ α = β ◦ φ). Moreover, φ is a pull-back morphism if and only if ψ is. (vi) A pull-back morphism φ ∈ Arr(C) is an FSM-morphism (respectively, ﬁberwise-surjective morphism; monomorphism; irreducible morphism) if and only if Base(φ) ∈ Arr(D) is. (vii) Let φ : A → B be a co-angular pre-step; : C → B a morphism. Then there exists a co-angular pre-step γ : D → C and a morphism α : D → A such that ◦ γ = φ ◦ α. In particular, every co-angular pre-step of C is an FSM-morphism. def Proof. First, we consider assertion (i). Let A, B ∈ Ob(C); A D = Base(A); def B = Base(B); φ D : A D → B D a morphism in D. By the equivalence of categories of Deﬁnition 1.3, (i), (c), it follows that there exists a pull-back morphism ψ : C → B def of C such that ψ D = Base(ψ) : C D → B D of D deﬁnes an object of D B D that is isomorphic to the object deﬁned by φ D . In particular, C D is isomorphic to A D . Since C is of base-trivial type, it thus follows that A, C are isomorphic, so we may assume that A = C. Thus, ψ projects to a morphism ψ D : A D → B D of D such that φ D = ψ D ◦ δ, for some δ ∈ Aut D (A D ). Since C is of Aut-ample type, it thus follows that δ lifts to a γ ∈ Aut C (A). Thus, taking ψ ◦ γ : A → B yields a morphism of C that projects to φ D . This completes the proof of assertion (i). def def Next, we consider assertion (ii). Let A, B ∈ Ob(C); A D = Base(A); B D = Base(B); φ, ψ : A → B pull-back morphisms of C that project to the same morphism A D → B D of D. By the deﬁnition of a “pull-back morphism” [cf. Deﬁnition 1.2, (ii)], it thus follows formally that there exist base-identity endomorphisms α, β ∈ End C (A) such that ψ = φ ◦ α, φ = ψ ◦ β. In particular, we obtain that ψ = ψ ◦ β ◦ α, φ = φ ◦ α ◦ β, hence [again by Deﬁnition 1.2, (ii)] that α ◦ β, β ◦ α are both equal to the identity endomorphism of A, i.e., that α, β ∈ Aut C (A). But this implies that α, β ∈ O × (A) = {1}, so φ = ψ, as desired. This completes the proof of assertion (ii). Next, we consider assertion (iii). The existence and uniqueness of β as asserted follows immediately from the isomorphism of functors appearing in the deﬁnition of a “pull-back morphism” [cf. Deﬁnition 1.2, (ii)]. This completes the proof of assertion (iii). Now since a co-angular linear morphism factors as the composite of a pull-back morphism with a co-angular pre-step [cf. Propositions 1.4, (iv); 1.7, (iii)], the existence of the map “→” of assertion (iv) follows immediately [cf. Proposition 1.7, (iii)] from assertion (iii) and Deﬁnition 1.3, (iii), (c); the asserted injectivity of this map follows from the total epimorphicity of C; the fact that this map is uniquely determined by the condition given in assertion (iii) follows from the fact that pre-steps are monomorphisms [cf. Deﬁnition 1.3, (v), (a)], and the deﬁnition of a “pull-back morphism” in Deﬁnition 1.2, (ii). 38 SHINICHI MOCHIZUKI Next, we consider assertion (v). First, we observe that the uniqueness of ψ follows from the fact that β is a monomorphism [cf. Deﬁnition 1.3, (v), (a)] in the non-resp’d case and from the total epimorphicity of C applied to α in the resp’d case. When φ is a pull-back morphism [hence co-angular and linear — cf. Proposition 1.4, (ii)], the existence of a pull-back morphism ψ as desired follows immediately by applying the equivalence of categories induced by the projection functor in Def- inition 1.3, (i), (c); the deﬁnition of a “pull-back morphism” in Deﬁnition 1.2, (ii); Proposition 1.7, (i), (v) [applied to co-angular linear morphisms]; and the equiva- lences of categories of Deﬁnition 1.3, (iii), (d). When φ is an isometric pre-step, the existence of an isometric pre-step ψ as desired follows immediately from the equiv- alence of categories of Proposition 1.9, (ii) [in the “case of a co-angular pre-step”]. When φ is a co-angular pre-step, the existence of a co-angular pre-step ψ as desired follows formally from the equivalences of categories of Deﬁnition 1.3, (iii), (d). In light of the factorizations of Deﬁnition 1.3, (v), (b), (c); Proposition 1.7, (iii), this completes the proof of assertion (v). Next, we observe that assertion (vi) follows formally from the isomorphism of functors appearing in the deﬁnition of a “pull-back morphism” [cf. Deﬁnition 1.2, (ii)], together with the equivalence of categories induced by the projection functor in Deﬁnition 1.3, (i), (c) [cf. also Proposition 1.7, (v), for pull-back morphisms]. Finally, we consider assertion (vii). By applying the factorizations of Deﬁnition 1.3, (iv), (a); Deﬁnition 1.3, (v), (b), it follows immediately that we may assume without loss of generality [from the point of view of showing the existence of γ, α with the desired properties] that is a pull-back morphism, an isometric pre- step, a co-angular pre-step, or a morphism of Frobenius type. If is a pull-back morphism, then it follows immediately [by “pulling back the zero divisor of φ via ” — cf. assertion (v)] that there exist a pull-back morphism α : D → A and a co-angular pre-step γ : D → C such that ◦ γ = φ ◦ α. Next, observe that if is an isometric pre-step, then the existence of γ, α with the desired properties follows formally from the equivalence of categories of Proposition 1.9, (ii) [induced by φ]. Next, observe that if is a co-angular pre-step, then it follows immediately from the second equivalence of categories of Deﬁnition 1.3, (iii), (d), that there exist co-angular pre-steps α : D → A, γ : D → C such that ◦ γ = φ ◦ α. Finally, we consider the case where is a morphism of Frobenius type. By applying the second equivalence of categories of Deﬁnition 1.3, (iii), (d), it follows that we may assume [by replacing φ by the composite of φ with an appropriate pre-step A → A] that Div(φ) = deg Fr ( ) · x, for some x ∈ Φ(A). Thus, [by applying again the second equivalence of categories of Deﬁnition 1.3, (iii), (d)], it follows that there exist a morphism of Frobenius type α : D → A and a co-angular pre-step γ : D → C such that ◦ γ = φ ◦ α [cf. also Proposition 1.10, (i)]. This completes the proof of the existence of γ, α with the desired properties. It thus follows formally that every co-angular pre-step of C is ﬁberwise surjective. On the other hand, by Deﬁnition 1.3, (v), (a), every pre-step is a monomorphism. Thus, we conclude that every co- angular pre-step of C is an FSM-morphism. This completes the proof of assertion (vii). Remark 1.11.1. Observe that in the situation of Proposition 1.11, (iii), if α is THE GEOMETRY OF FROBENIOIDS I 39 a morphism of Frobenius type, and β D is an isomorphism, then β is a morphism of Frobenius type. [Indeed, then β is co-angular by Deﬁnition 1.3, (iii), (b), and isometric by Remark 1.1.1.] In particular, it follows [cf. Remark 1.2.1] that [at least in the case of Frobenioids] “Frobenius-trivial” implies “universally Div-Frobenius- trivial”. Proposition 1.12. (Endomorphisms) Let Φ be a divisorial monoid on a connected, totally epimorphic category D; C → F Φ a Frobenioid; A ∈ Ob(C); def A D = Base(A) ∈ Ob(D). Then: (i) We have natural exact sequences of monoids 1 → O × (A) → Aut C (A) → Aut D (A D ) 1 → O (A) → End C (A) → N ≥1 × End D (A D ) — where the second arrow in each sequence is the natural inclusion; the third ar- row of the ﬁrst sequence is determined by the natural projection functor to D; the third arrow of the second sequence is determined by the Frobenius degree and the natural projection functor to D. If, moreover, A is Aut-ample (respectively, End- ample; quasi-Frobenius-trivial), then the map Aut C (A) → Aut D (A D ) (respec- tively, End C (A) → End D (A D ); End C (A) → N ≥1 ) is surjective. (ii) An endomorphism of A is a sub-automorphism [cf. §0] if and only if it is an isometric linear endomorphism that projects to a sub-automorphism of D. (iii) A sub-automorphism of A is an automorphism if and only if it is a base- isomorphism. (iv) Suppose that A is Aut sub -ample. Then A is Aut-saturated [cf. §0] if and only if A D is. Proof. Assertion (i) is immediate from the deﬁnitions. The necessity of the condi- tions of assertion (ii), (iii) is immediate from Remark 1.1.1. To prove the suﬃciency of the conditions of assertion (ii), (iii), it suﬃces, in light of the equivalence of cate- gories [involving pull-back morphisms] of Deﬁnition 1.3, (i), (c) [cf. also Proposition 1.11, (iii)], and the fact that endomorphisms are always co-angular [cf. Deﬁnition 1.3, (iii), (b)], to observe that any LB-invertible linear base-isomorphism [i.e., LB- invertible pre-step] is, in fact, an isomorphism [cf. Proposition 1.4, (iii)]. Now assertion (iv) follows formally from assertions (ii), (iii) and the deﬁnitions. Proposition 1.13. (Rigidity and Slimness) Let Φ be a divisorial monoid on a connected, totally epimorphic category D; C → F Φ a Frobenioid; A ∈ Ob(C); def A D = Base(A) ∈ Ob(D). Suppose further that the category D is slim [cf. §0]. Then: 40 SHINICHI MOCHIZUKI (i) The composite C A → D of the natural functor C A → C with the natural projection functor C → D is rigid [cf. §0]. In particular, the functor C → D is rigid. (ii) The composite C A → F Φ of the natural functor C A → C with the functor C → F Φ is rigid. In particular, the functor C → F Φ is rigid. (iii) Suppose, moreover, that every object A ∈ Ob(C) satisﬁes [at least] one of the following two conditions: (a) O × (A) imtr-pre = {1} [cf. Proposition 1.9, (ii)]; (b) n∈N ≥1 {O × (A)} n = {1}, and, moreover, there exists a co-angular pre-step ∼ B → A [which, by Deﬁnition 1.3, (iii), (c), induces a bijection O × (B) → O × (A)] such that B is quasi-Frobenius-trivial and Frobenius-normalized. Then the category C is slim. Proof. First, we consider assertion (i). Any automorphism α of the functor C A → pl-bk → C A → D. On D determines an automorphism of the composite functor C A pl-bk → D A D → D, the other hand, this composite functor factors as a composite C A pl-bk where the ﬁrst functor C A → D A D is [by Deﬁnition 1.3, (i), (c)] an equivalence of categories. Thus, we conclude that α determines an automorphism of the natural functor D A D → D, which is necessarily trivial, since D is slim. Since A is arbitrary, we thus conclude that both C A → D and C → D are rigid. This completes the proof of assertion (i). Next, we consider assertion (ii). Let α be an automorphism of the functor C A → F Φ . By assertion (i), it follows that the automorphisms of objects of F Φ [which, by Proposition 1.5, (i), is itself a Frobenioid] induced by α are base-identity automorphisms. Since Φ is divisorial, hence, in particular, sharp [cf. Deﬁnition 1.1, (i), (ii)], it thus follows that all of these automorphisms are trivial, hence that α is trivial. Since A is arbitrary, we thus conclude that both C A → F Φ and C → F Φ are rigid. This completes the proof of assertion (ii). Finally, we consider assertion (iii). Let α be an automorphism of the natural functor C A → C. By assertion (i), it follows that the automorphisms of objects of C induced by α are base-identity automorphisms, i.e., belong to “O × (−)”. Moreover, the functoriality of the automorphisms induced by α with respect to isometric pre- steps implies that these automorphisms belong to “O × (−) imtr-pre ”. Similarly, the functoriality of the automorphisms induced by α with respect to base-identity en- domorphisms implies that, at least in the case of quasi-Frobenius-trivial, Frobenius- normalized objects — hence also [cf. Deﬁnition 1.3, (iii), (c)] objects as in (b) of the statement of assertion (iii) — these automorphisms belong to “ n∈N ≥1 {O × (−)} n ”. Thus, we conclude that under either of the assumptions (a), (b) in the statement of assertion (iii), the automorphisms induced by α are trivial. This completes the proof of assertion (iii). Remark 1.13.1. Note that if the hypothesis of Proposition 1.13, (iii), fails to hold, then it is not necessarily the case that C is slim. Indeed, if M is a perfect THE GEOMETRY OF FROBENIOIDS I 41 pre-divisorial monoid, and C is a one-object category whose unique object has endo- morphism monoid equal to the elementary Frobenioid F M [so C equipped with the functor of one-object categories determined by the natural morphism of monoids F M → F M char is a Frobenioid, by Proposition 1.5, (i)], then any collection of ele- ments {α n } n∈N ≥1 of M ± such that α nm = m · α n for all n, m ∈ N ≥1 determines an automorphism of the natural functor C A → C [which is nontrivial as soon as any of the α n is nonzero] by assigning to an arrow φ : B → A of C the automorphism α deg Fr (φ) ∈ Aut C (B). One key result for analyzing the category-theoretic structure of Frobenioids [cf. §3] is the following: Proposition 1.14. (Irreducible Morphisms) Let Φ be a divisorial monoid on a connected, totally epimorphic category D; C → F Φ a Frobenioid of isotropic type; φ ∈ Arr(C). Suppose further that D is of FSMFF-type [cf. §0]. Then: (i) φ is irreducible if and only if φ is one of the following: (a) a prime- Frobenius morphism; (b) a step such that Div(φ) is irreducible; (c) a pull-back morphism such that Base(φ) is an irreducible morphism of D. (ii) φ is a pre-step if and only if it is an FSM-morphism that is mid- adjoint [cf. §0] to the irreducible morphisms which are not pre-steps. (iii) Suppose that φ is irreducible. Then φ is a non-pre-step if and only if the following condition holds: There exists an N ∈ N ≥1 such that for every equality of composites in C α n ◦ α n−1 ◦ . . . ◦ α 2 ◦ α 1 = ψ ◦ φ — where α 1 , . . . , α n , ψ are FSMI-morphisms [cf. §0] — it holds that n ≤ N . (iv) Let α ◦ β = β ◦ α be an equality of composites of C, where deg Fr (β) = deg Fr (β ), and α, α are irreducible. Then α is a prime-Frobenius morphism if and only if α is; moreover, deg Fr (α) = deg Fr (α ). (v) Suppose further that Φ is non-dilating, and that φ is a non-pre-step irreducible endomorphism of a non-group-like object A ∈ Ob(C). Then φ is a Div-identity prime-Frobenius endomorphism if and only if the following condition holds: For every step α : A → B, there exists a non-pre-step irreducible morphism ψ : B → B and a step β : B → B such that ψ ◦ α = β ◦ α ◦ φ. Proof. First, we consider assertion (i). The suﬃciency of the condition of assertion (i) follows for morphisms as in (a) (respectively, (b); (c)) from Proposition 1.10, (iv) (respectively, the equivalences of categories of Deﬁnition 1.3, (iii), (d) [cf. also Propositions 1.4, (i); 1.7, (v)]; Proposition 1.11, (vi)). To verify the necessity of the condition of assertion (i), observe that it follows formally from the factorization of Deﬁnition 1.3, (iv), (a), that φ is either a morphism of Frobenius type, a step, or a pull-back morphism. Thus, by Propositions 1.7, (v); 1.10, (iv); 1.11, (vi), the irreducibility of φ implies immediately that φ is a morphism as in (a), (b), or (c). 42 SHINICHI MOCHIZUKI Next, we consider assertion (ii). To verify the suﬃciency of the condition of assertion (ii), observe ﬁrst that by the factorization of Deﬁnition 1.3, (iv), (a), we may write φ = α ◦ β ◦ γ, where α is a pull-back morphism, β is a pre-step, and γ is a morphism of Frobenius type. By assertion (i) [cf. also Proposition 1.10, (v)], it follows that γ is an isomorphism; thus, we may assume without loss of generality that γ is the identity, i.e., φ = α ◦ β. On the other hand, it follows formally from the fact that φ is an FSM-morphism that α is ﬁberwise-surjective [cf. §0]. Next, I claim that α is a monomorphism. Indeed, write φ : A → B, β : A → C, α : C → B; let 1 , 2 : D → C be such that α ◦ 1 = α ◦ 2 . Then by Remark 1.1.1, it follows immediately that deg Fr ( 1 ) = deg Fr ( 2 ), Div( 1 ) = Div( 2 ), hence, by applying the factorization of Deﬁnition 1.3, (iv), (a) [and the total epimorphicity of C; cf. also Deﬁnition 1.3, (ii), and the equivalences of categories of Deﬁnition 1.3, (iii), (d)], we may assume without loss of generality [from the point of view of showing that α is a monomorphism] that 1 , 2 are pull-back morphisms. Now by “adding the pull-backs of β ∗ (Div(β)) via 1 , 2 ” [cf. Proposition 1.11, (v); the equivalences of categories of Deﬁnition 1.3, (iii), (d)], it follows that there exists a pre-step ζ : E → D such that there exist γ 1 , γ 2 ∈ Arr(C) satisfying 1 ◦ ζ = β ◦ γ 1 , 2 ◦ ζ = β ◦ γ 2 . Thus, we have: φ ◦ γ 1 = α ◦ β ◦ γ 1 = α ◦ 1 ◦ ζ = α ◦ 2 ◦ ζ = α ◦ β ◦ γ 2 = φ ◦ γ 2 . But since φ is [an FSM-morphism, hence, in particular] a monomorphism, it follows that γ 1 = γ 2 , hence [by the total epimorphicity of C] that 1 = 2 . This completes the proof of the claim. In particular, we conclude that α is an FSM-morphism. Thus, it follows [cf. Proposition 1.11, (vi)] that Base(α) is an FSM-morphism of D. Since, however, we are operating under the assumption that D is of FSMFF- type, it follows that if α is not an isomorphism, then Base(α) admits a subordinate [cf. condition (a) of the deﬁnition of a “category of FSMFF-type” in §0] FSMI- morphism, which implies [cf. Proposition 1.11, (vi)] that α admits a subordinate FSMI-morphism [which is also a pull-back morphism]. Since φ, however, is as- sumed to be mid-adjoint to the irreducible morphisms which are not pre-steps, we thus obtain a contradiction. Thus, α is an isomorphism, so φ is a pre-step. This completes the proof of the suﬃciency of the condition of assertion (ii). Next, we consider the necessity of the condition of assertion (ii). Thus, suppose that φ is a pre-step. By Proposition 1.11, (vii), φ is an FSM-morphism; by Proposition 1.7, (v), φ is mid-adjoint to the non-pre-steps. This completes the proof of assertion (ii). Next, we consider assertion (iii). By assertion (i), it suﬃces to show that assertion (iii) holds for each of the three types of morphisms “(a), (b), (c)” discussed in assertion (i). If φ is an irreducible pre-step, then it follows immediately — by taking ψ to be a prime-Frobenius morphism of increasingly large Frobenius degree [cf. Proposition 1.10, (ii)] — that the condition in the statement of assertion (iii) is false [as desired]. On the other hand, if φ is a non-pre-step, then it is an isometry. Now if the condition in the statement of assertion (iii) is false, then there exist equalities α n ◦ α n−1 ◦ . . . ◦ α 2 ◦ α 1 = ψ ◦ φ where α 1 , . . . , α n , ψ are FSMI-morphisms, and n is arbitrarily large. Here, we note that since ψ ◦φ and ψ are FSM-morphisms, it thus follows formally that φ is also an FSM-morphism. Next, observe that since φ is an isometry, it follows from the fact THE GEOMETRY OF FROBENIOIDS I 43 that ψ is irreducible [cf. also assertion (i); Deﬁnition 1.1, (ii), (b); Remark 1.1.1] that Div(ψ ◦ φ) is either zero or irreducible; since, moreover, deg Fr (ψ ◦ φ) always divides a product of two prime numbers [cf. assertion (i); the irreducibility of φ, ψ], it thus follows that in any factorization of ψ ◦ φ by FSMI-morphisms, all but three [i.e., corresponding to two possible prime factors of the Frobenius degree, plus one possible irreducible factor of the zero divisor] of the factorizing FSMI-morphisms are pull-back morphisms [cf. assertion (i)]. On the other hand, this implies that factorizations of arbitrarily large length determine chains of FSMI-morphisms [cf. assertion (i); Proposition 1.11, (vi)] originating from the projection to D of the domain of φ which are also of arbitrarily large length, a contradiction [cf. condition (b) of the deﬁnition of a “category of FSMFF-type” in §0]. This completes the proof of assertion (iii). Next, we consider assertion (iv). Since deg Fr (β) = deg Fr (β ), it follows from Remark 1.1.1 that deg Fr (α) = deg Fr (α ), hence [since α, α are irreducible], by assertion (i), that α is a prime-Frobenius morphism if and only if α is. This completes the proof of assertion (iv). Finally, we consider assertion (v). First, we observe that the necessity of the condition in the statement of assertion (v) [where we take ψ to be a prime-Frobenius morphism such that deg Fr (φ) = deg Fr (ψ)] follows immediately from Proposition 1.10, (i) [cf. also Deﬁnition 1.3, (ii); assertion (i); the ﬁrst equivalence of categories of Deﬁnition 1.3, (iii), (d)]. Next, we consider suﬃciency. To show that φ is a prime-Frobenius morphism, it suﬃces [by assertion (i)] to show that it is not a pull-back morphism. Thus, suppose that φ is a pull-back morphism. Since A is non- group-like, it follows [cf. Proposition 1.4, (iii)] that there exists a step α : A → B, hence that there exist ψ, β as in the statement of assertion (v). By assertions (i), def def (iv), ψ is also a pull-back morphism. Write x = Div(α), y = α ∗ (Div(β)) [where, for simplicity, we write α ∗ for Φ(Base(α))]. Then by Remark 1.1.1, it follows that φ ∗ (x + y) = x, i.e., that φ ∗ (x) ≤ x. Since x = 0 is arbitrary [cf. the ﬁrst equivalence of categories of Deﬁnition 1.3, (iii), (d)], it thus follows from our assumption that Φ is non-dilating that φ ∗ is the identity morphism. But this implies that x + y = x, i.e., [since Φ is integral — cf. Deﬁnition 1.1, (i)] that y = 0, in contradiction to our assumption that β is a step [i.e., not just a [necessarily co-angular!] pre-step — cf. Proposition 1.4, (i), (iii)]. Thus, we conclude [cf. assertion (iv)] that φ, ψ are prime-Frobenius morphisms, of the same Frobenius degree. In particular, if def def we write x = Div(α), y = α ∗ (Div(β)), then it follows [cf. Remark 1.1.1] that φ ∗ (x + y) = deg Fr (φ) · x, i.e., that φ ∗ (x) x [cf. §0], hence [by our assumption that Φ is non-dilating] that φ ∗ is the identity morphism. This completes the proof of assertion (v). 44 SHINICHI MOCHIZUKI Section 2: Frobenius Functors In the present §2, we discuss various functors between Frobenioids that are intended to be reminiscent of the Frobenius morphism in positive characteristic scheme theory. In the following discussion, we maintain the notation of §1. Also, we assume that we have been given a divisorial monoid Φ on a connected, totally epimorphic category D and a Frobenioid C → F Φ . Proposition 2.1. (The Naive Frobenius Functor) Let d ∈ N ≥1 . Then: (i) The assignment A → A ; φ → φ — where φ : A → B is an arbitrary morphism of C; α : A → A , β : B → B are morphisms of Frobenius type of Frobenius degree d; φ is the unique morphism such that φ ◦ α = β ◦ φ [cf. Proposition 1.10, (i)] — determines a functor Ψ: C→C [well-deﬁned up to isomorphism of functors] which we shall refer to as the naive Frobenius functor [of degree d] on C. Finally, the composite of the naive Frobe- nius functor of degree d 1 ∈ N ≥1 on C with the naive Frobenius functor of degree d 2 ∈ N ≥1 on C is isomorphic to the naive Frobenius functor of degree d 1 · d 2 on C. (ii) The functor Ψ of (i) is “1-compatible”, relative to C → F Φ , with the functor F Φ → F Φ — which we shall refer to as the Frobenius functor on F Φ — determined [cf. Deﬁnition 1.1, (iii)] by the endomorphism of the functor Φ given by multiplication by d. Moreover, if, in the notation of (i), A = A , A is Frobenius-normalized, and the morphism α : A → A is taken to be a base- identity endomorphism, then the morphism of monoids O (A) → O (A ) induced by Ψ is given by raising to the d-th power. (iii) C is of perfect type if and only if Ψ is an equivalence of categories. Proof. Assertions (i), (ii) follow immediately from Deﬁnition 1.3, (ii); Proposition 1.10, (i) [cf. also Proposition 1.7, (i)]. Finally, we consider assertion (iii). The suﬃciency of the condition of assertion (iii) follows immediately from the deﬁnition of “perfect” [cf. Deﬁnition 1.2, (iv); Remark 1.1.1]. To verify necessity, suppose that C is of perfect type. Then the essential surjectivity of Ψ follows immediately from the deﬁnition of “perfect” [cf. Deﬁnition 1.2, (iv)]. To verify that Ψ is fully faithful, we reason as follows: In light of the 1-compatibility of Ψ with the Frobenius functor on F Φ [cf. assertion (ii)], the total epimorphicity of C, and the factorization of Deﬁnition 1.3, (iv), (a), it follows immediately that one may reduce to the case of linear morphisms by applying the existence and (essential) uniqueness of morphisms of Frobenius type of a given Frobenius degree [cf. Deﬁnition 1.3, (ii)]. Moreover, THE GEOMETRY OF FROBENIOIDS I 45 by applying the equivalence of categories [involving pull-backs] of Deﬁnition 1.3, (i), (c) [cf. also the isomorphism of functors appearing in the deﬁnition of a “pull-back morphism” in Deﬁnition 1.2, (ii)], one may reduce further to the case of pre-steps. But the case of pre-steps follows immediately from the deﬁnition of “perfect” [cf. Deﬁnition 1.2, (iv)]. This completes the proof of assertion (iii). Remark 2.1.1. If C is of perfect type, then for any d = a/b ∈ Q >0 , where a, b ∈ N ≥1 , composing the naive Frobenius functor of degree a with some quasi- inverse functor to the naive Frobenius functor of degree b yields a “naive Frobenius functor of degree d”, which, by Proposition 2.1, (i), is independent of the choice of a, b. Proposition 2.2. (The Functor O (−)) Write D ∗ for the category whose objects are the objects of C istr and whose morphisms are given as follows: def Hom D ∗ (A, B) = Hom D (A D , B D ) def def [where A, B ∈ Ob(C istr ); A D = Base(A); B D = Base(B)]. Thus, the natural projection functor C → D determines natural functors C istr → D ∗ → D. Moreover: (i) The functor D ∗ → D is an equivalence of categories. (ii) There is a unique contravariant functor D ∗ → Mon Ob(C istr ) = Ob(D ∗ ) A → O (A) ∈ Ob(Mon) such that for φ : A → B in Arr(C istr ), with image φ D ∗ in D ∗ , the following properties are satisﬁed: (a) if φ is a [necessarily co-angular – cf. Proposition 1.4, (i)] linear morphism, then O (φ D ∗ ) : O (B) → O (A) is the inclusion of Proposition 1.11, (iv); (b) if φ is a [necessarily co-angular] pre-step, then O (φ D ∗ ) : O (B) → O (A) is the bijection of Deﬁnition 1.3, (iii), (c). By abuse of notation, we shall also denote by “O (−)” the restriction of this functor on D ∗ to (C istr ) lin . Finally, by applying the equivalence of categories of (i), we obtain a contravariant functor D → Mon, which, by abuse of notation, we shall also denote by “O (−)”, and which is well-deﬁned up to isomorphism. (iii) The assignment Ob(C istr ) A → O × (A) (⊆ O (A)) determines a sub- functor of the functor of (ii) which is equal to the subfunctor A → O (A) ± [cf. the notation of §0]. Moreover, the operation “Div(−)” determines a functorial homomorphism O (A) → Φ(A) that induces an inclusion O (A) char = O (A)/O × (A) → Φ(A) [cf. the notation of §0]. (iv) If φ : A → A istr is an isotropic hull in C, then φ determines a natural inclusion of monoids O (A) → O (A istr ). 46 SHINICHI MOCHIZUKI Proof. As for assertion (i), essential surjectivity follows immediately from Deﬁni- tion 1.3, (i), (a) [i.e., applied to the Frobenioid C istr — cf. Proposition 1.9, (v)], while fully faithfulness follows formally from the deﬁnition of the category D ∗ . Next, def def we consider assertion (ii). Let A, B ∈ Ob(C istr ); A D = Base(A); B D = Base(B). Now observe that any morphism A D → B D in D factors as the composite of an ∼ def isomorphism A D → C D , where C ∈ Ob(C istr ), C D = Base(C), with a morphism C D → B D which is the projection to D of a pull-back morphism C → B of C istr [cf. Deﬁnition 1.3, (i), (c)]; moreover, this pull-back morphism is uniquely determined, istr as an object of C B , up to isomorphism [cf. the isomorphism of functors appearing in the deﬁnition of a “pull-back morphism” in Deﬁnition 1.2, (ii)]. Thus, it follows that to construct the desired functor “O (−)” on D ∗ , it suﬃces to construct, for ∼ each isomorphism φ D : A D → B D , a bijection O (φ D ) : O (A) → O (B) which is compatible with composition of isomorphisms. [Indeed, once one constructs “O (−)” in this fashion, the fact that this “O (−)” is compatible with composites of morphisms of D ∗ follows immediately from the manifest functoriality of the in- clusion of Proposition 1.11, (iv).] This may be done by using co-angular pre-steps γ A : C → A, γ B : C → B such that φ D = Base(γ B ) ◦ Base(γ A ) −1 [cf. Deﬁnition 1.3, ∼ ∼ (i), (b)] and the bijections O (γ A ) : O (A) → O (C), O (γ B ) : O (B) → O (C) determined by γ A , γ B [cf. Deﬁnition 1.3, (iii), (c)]. Note, moreover, that the result- ing bijection O (γ A ) −1 ◦ O (γ B ) is independent of the choice of γ A , γ B . [Indeed, if δ A : D → A, δ B : D → B satisfy φ D = Base(δ B ) ◦ Base(δ A ) −1 , then there exist [cf. Deﬁnition 1.3, (i), (b)] co-angular pre-steps C : E → C, D : E → D such that Base(γ A ) ◦ Base( C ) = Base(δ A ) ◦ Base( D ) — which [since Base(δ B ) ◦ Base(δ A ) −1 = Base(γ B ) ◦ Base(γ A ) −1 ] implies that Base(γ B ) ◦ Base( C ) = Base(δ B ) ◦ Base( D ) hence that O (γ A ◦ C ) = O (δ A ◦ D ); O (γ B ◦ C ) = O (δ B ◦ D ) [cf. Deﬁnition 1.3, (iii), (c)], i.e., that O ( C ) ◦ O (γ A ) = O ( D ) ◦ O (δ A ); O ( C ) ◦ O (γ B ) = O ( D ) ◦ O (δ B ) — that is to say O (γ A ) −1 ◦ O (γ B ) = O (δ A ) −1 ◦ O (δ B ) as desired.] This completes the proof of assertion (ii). Assertion (iii) is immediate from the deﬁnitions [cf. also Deﬁnition 1.3, (iii), (b); Deﬁnition 1.3, (vi)]. Assertion (iv) follows immediately from the “universal property of an isotropic hull” [cf. Deﬁnition 1.2, (iv)] and the fact that an isotropic hull is always a monomorphism [cf. Deﬁnition 1.3, (v), (a)]. THE GEOMETRY OF FROBENIOIDS I Deﬁnition 2.3. in monoids 47 We shall refer to as a characteristic splitting on C a subfunctor τ : (C istr ) lin → Mon of the functor O (−) : (C istr ) lin → Mon of Proposition 2.2, (ii), such that the following properties hold: (a) for every A ∈ Ob(C istr ), τ (A) maps bijectively onto O (A) char , hence determines a splitting of monoids ∼ O × (A) × τ (A) → O (A) which is functorial in A; (b) for every isotropic hull A → A istr of C, τ (A istr ) ⊆ O (A istr ) lies in the image of O (A) via the natural injection of Proposition 2.2, (iv). Deﬁnition 2.4. (i) We shall say that M ∈ Ob(Mon) is perf-factorial if it satisﬁes the following conditions: (a) M is divisorial. (b) For every p ∈ Prime(M ) [cf. §0], the monoid M p is monoprime [cf. §0]. (c) The map def p∈Prime(M ) rlf M pf → M factor = M p rlf a → (. . . , sup(Bound p Ë {0} (a)), . . . ) def [where we write M p rlf = M p pf ⊗R ≥0 ; we refer to §0 for more on the notation “M pf ”, “M p pf ”, “⊗R ≥0 ”; the “sup” at the index p is taken in M p rlf ] is a well-deﬁned [i.e., the various Bound p Ë {0} (a) ⊆ M p rlf are bounded subsets] injective homomorphism of monoids whose image lies in p∈Prime(M ) M p pf , hence determines an injective homomorphism def pf = M pf → M factor M p pf p∈Prime(M ) which we shall refer to as the factorization homomorphism of M pf . We shall often use the factorization homomorphism to regard M pf as a sub- pf rlf ⊆ M factor . monoid of M factor rlf (d) If a ∈ M factor , then we shall write Supp(a) ⊆ Prime(M ) for the subset of p for which the component of a at p is nonzero and refer to Supp(a) as the pf support of a. Then the submonoid M pf ⊆ M factor satisﬁes the following pf pf property: If a ∈ M factor and b ∈ M satisfy Supp(a) ⊆ Supp(b), then a ∈ M pf . [Thus, in particular, if a, b ∈ M pf , then an inequality “a ≤ b” pf .] holds in M pf if and only if it holds in M factor 48 SHINICHI MOCHIZUKI Now suppose that M is perf-factorial. Then we shall refer to the [subset which is easily veriﬁed to be a] submonoid rlf M rlf ⊆ M factor rlf of elements a ∈ M factor such that there exists a b ∈ M pf satisfying Supp(a) ⊆ pf Supp(b) as the realiﬁcation of M . Thus, both the submonoid M pf ⊆ M factor and the rlf rlf submonoid M ⊆ M factor are completely determined by the collection of subsets Supp(a) ⊆ Prime(M ), as a ranges over the elements of M pf ; if a, b ∈ M rlf , then an rlf ; inequality “a ≤ b” holds in M rlf if and only if it holds in M factor rlf ) gp = (M rlf ) gp ⊆ (M factor (M p rlf ) gp p∈Prime(M ) is an R-vector space. Finally, one veriﬁes immediately that M pf , M rlf are also perf-factorial. (ii) Let Λ be a monoid type. Then we shall say that Λ supports M ∈ Ob(Mon) if any of the following conditions hold: (a) Λ = Z; (b) Λ = Q, and M is perfect; (c) Λ = R, M is perfect and perf-factorial, and for every p ∈ Prime(M ), the monoid M p is R-monoprime. Note that if Λ supports M , then Λ >0 acts naturally on M . (iii) Let Λ be a monoid type that supports Φ [cf. Deﬁnition 1.1, (ii)]; d ∈ Λ >0 . Then we shall write d · Φ(−) ⊆ Φ(−) for the subfunctor of Φ determined by the assignment Ob((C istr ) lin ) (Φ(A)) (⊆ Φ(A)) and C (d) ⊆ C A → d · for the subcategory determined by the arrows whose zero divisor lies in d · Φ(−) ⊆ Φ(−). Finally, multiplication by d on Φ(−) determines a “Frobenius functor” [as- sociated to d — cf. Proposition 2.1, (ii)] F Φ → F Φ which is compatible with Frobenius degrees and the natural projection functor F Φ → D. Proposition 2.5. (The Unit-linear Frobenius Functor) Let τ be a charac- teristic splitting on C; Λ a monoid type that supports Φ; d ∈ Λ >0 . Suppose that the Frobenioid C is of Frobenius-normalized, metrically trivial, and Aut- ample type. Then: (i) The natural inclusion O (A) char → Φ(A), where A ∈ Ob(C istr ), of Propo- sition 2.2, (iii), is, in fact, a bijection. (ii) C istr is of base-trivial type. Moreover, every object of C istr is Frobenius- trivial. THE GEOMETRY OF FROBENIOIDS I 49 (iii) There exists an equivalence of categories ∼ Ψ : C → C (d) — which we shall refer to as the unit-linear Frobenius functor [associated to τ , d] — that satisﬁes the following properties: (a) Ψ acts as the identity on objects and isometries of C; (b) Ψ is 1-compatible, relative to the functors C → F Φ , C (d) → F d·Φ = (F Φ ) (d) ⊆ F Φ with the Frobenius functor associated to d on F Φ [which implies, in particular, that C (d) , equipped with the natural functor C (d) → F d·Φ , is a Frobenioid]. Proof. First, we observe that, by applying either of the equivalences of categories of Deﬁnition 1.3, (iii), (d), assertion (i) follows formally from the fact that C is of metrically trivial and Aut-ample type. Next, we consider assertion (ii). Since C is of metrically trivial type, it follows from the existence of [necessarily co-angular — cf. Proposition 1.4, (i)] pre-steps relating base-isomorphic objects of C istr [cf. Deﬁnition 1.3, (i), (b)], that the isomorphism class of an object of C istr is completely determined by the isomorphism class of D to which it projects. In particular, it follows from the existence of Frobenius-trivial objects [cf. Deﬁnition 1.3, (i), (a)] that every object of C istr is Frobenius-trivial. This completes the proof of assertion (ii). Finally, we consider assertion (iii). By applying the factorizations of Deﬁnition 1.3, (iv), (a); (v), (c), together with the bijection of assertion (i) [cf. also the equiv- alences of categories of Deﬁnition 1.3, (iii), (d)], we conclude that every morphism φ of C admits a factorization φ = α ◦ β ◦ γ ◦ δ in C, where α is a pull-back morphism; β is a base-identity pre-step endomorphism [hence is co-angular, by Deﬁnition 1.3, (iii), (b)]; γ is an isometric pre-step; δ is a morphism of Frobenius type. Moreover, this factorization is unique [cf. Deﬁnition 1.3, (iv), (a); (v), (c)], up to replacing (α, β, γ, δ) by (α◦ , −1 ◦β◦ζ, ζ −1 ◦γ◦θ, θ −1 ◦δ), where , θ are isomorphisms of C, and ζ = β ◦ , for some base-identity automorphism β . Suppose that β ∈ O (A), for A ∈ Ob(C). Thus, by applying the characteristic splitting ∼ O × (A) × τ (A) → O (A) [which applies even if A is not isotropic — cf. Deﬁnition 2.3, (a), (b)] to β ∈ O (A), we obtain a factorization β = β 0 · β 1 [where β 0 ∈ O × (A), β 1 ∈ τ (A)]. Now we set def Ψ(β) = β 0 · β 1 d ; def Ψ(φ) = α ◦ Ψ(β) ◦ γ ◦ δ 50 SHINICHI MOCHIZUKI [where we note that the expression “β 1 d ” makes sense for d ∈ Λ >0 , by assertion (i); Deﬁnition 2.4, (ii)]. Then it follows immediately from the functoriality of the ∼ characteristic splitting τ (−) that for any isomorphism : A → A in C, β ∈ O × (A), we have Ψ( −1 ◦ β ◦ β ◦ ) = −1 ◦ Ψ(β) ◦ β ◦ . This implies immediately that Ψ(φ) is independent of the choice of factorization φ = α ◦ β ◦ γ ◦ δ. Next, observe that by assertion (ii), it follows that if φ ∈ Arr(C istr ), then the morphism of Frobenius type δ may be taken to be a base-identity endomorphism. Thus, by the functoriality of τ with respect to morphisms of (C istr ) lin , and our assumption that C is of Frobenius-normalized type — together with the elementary computation Ψ(β d ) = Ψ(β 0 d · β 1 d ) = β 0 d · (β 1 d ) d = (β 0 · β 1 d ) d = Ψ(β) d for d ∈ N ≥1 — it follows that the assignment φ → Ψ(φ) is compatible with com- posites, at least when φ ∈ Arr(C istr ). On the other hand, since isotropic hulls are monomorphisms [cf. Deﬁnition 1.3, (v), (a)], this implies [by relating an arbitrary φ ∈ Arr(C) to the result of applying the isotropiﬁcation functor of Proposition 1.9, (v), to φ] that the assignment φ → Ψ(φ) is compatible with composites, for arbitrary φ ∈ Arr(C). This completes the deﬁnition of a functor Ψ : C → C (d) which satisﬁes the properties (a), (b) in the statement of Proposition 2.5, (iii). On the other hand, it is clear from the deﬁnition of Ψ, C (d) that Ψ is essentially surjective, faithful, and full [cf. assertion (i)]. This completes the proof of Proposition 2.5. Remark 2.5.1. If C is of isotropic and unit-trivial type, then the “unit-linear Frobenius functor” of Proposition 2.5, (iii), may be regarded as a sort of gener- alization of the “naive Frobenius functor” of Proposition 2.1, (i), to the case of d ∈ N ≥1 . Corollary 2.6. (Unit-wise Frobenius Functors I) Let τ be a character- istic splitting on C; d ∈ N ≥1 . Suppose that the Frobenioid C is of Frobenius- normalized, metrically trivial, and Aut-ample type. Then there exists a func- tor Ψ: C→C — which we shall refer to as the unit-wise Frobenius functor [associated to τ , d] — which satisﬁes the following properties: (a) Ψ is 1-compatible, relative to the functor C → F Φ , with the identity functor on F Φ . (b) Ψ maps an object (respectively, morphism of Frobenius type; pre-step; pull- back morphism) of C istr to an isomorphic object (respectively, abstractly equiva- lent morphism; abstractly equivalent morphism; abstractly equivalent mor- phism) of C. (c) If A ∈ Ob(C istr ), then there exists an isomorphism Ψ(A) ∼ = A such that the endomorphism of O × (A) induced by Ψ followed by conjugation by this isomorphism is given by raising to the d-th power. THE GEOMETRY OF FROBENIOIDS I 51 (d) If C is of perfect type, then Ψ is an equivalence of categories. If d = 1 or C is of isotropic and unit-trivial type, then Ψ is isomorphic to the identity functor. Proof. First, let us observe that the naive Frobenius functor C → C associated to d [cf. Proposition 2.1, (i)] factors naturally through the subcategory C (d) ⊆ C [cf. Proposition 2.1, (ii); Deﬁnition 2.4, (iii)]; write Ψ 1 : C → C (d) for the resulting functor. Next, let us write Ψ 2 : C (d) → C for some quasi-inverse functor to the unit-linear Frobenius functor [which is an equivalence of categories] associated def to d [cf. Proposition 2.5, (iii)]. Set Ψ = Ψ 2 ◦ Ψ 1 : C → C. Then it follows immediately from Propositions 2.1, (ii); 2.5, (iii), (b), that Ψ satisﬁes property (a). Since [cf. Proposition 2.5, (ii)] the isomorphism class of an object of C istr is completely determined by the isomorphism class of D to which it projects, it thus follows that Ψ preserves isomorphism classes of objects of C istr . Now the remainder of properties (b), (c), (d) follows immediately from the construction of Ψ 1 , Ψ 2 in the proofs of Propositions 2.1, (i); 2.5, (iii) [cf. also Remark 2.5.1; Proposition 1.10, (i); Deﬁnition 1.3, (ii); equivalences of categories of Deﬁnition 1.3, (iii), (d); the deﬁnition of a “pull-back morphism” in Deﬁnition 1.2, (ii); Proposition 2.1, (iii)]. Deﬁnition 2.7. Suppose that the Frobenioid C is of isotropic type. (i) We shall refer to as a base-section of the Frobenioid C any subcategory P ⊆ C pl-bk ⊆ C [where C pl-bk ⊆ C is as in Deﬁnition 1.3, (i), (c)] satisfying the following conditions: (a) P is a skeleton [cf. §0]; (b) every object of P is Frobenius- trivial; (c) the composite P → D of the inclusion functor P → C with the natural projection functor C → D is an equivalence of categories. In this situation, we shall refer to the morphisms of C that lie in P as P-distinguished. (ii) Let P ⊆ C be a base-section. Observe that since D, hence also P, is a connected category, it follows immediately that for any ∈ End(P → C), it makes sense to speak of the Frobenius degree deg Fr ( ) ∈ N ≥1 of — i.e., the Frobenius degree of the endomorphisms in C [of objects of P] determined by [which, since P is connected, is easily seen to be independent of the choice of object of P — cf. Remark 1.1.1]. We shall refer to as a [P-]Frobenius-section of the Frobenioid C any homomorphism of monoids F : N ≥1 → End(P → C) satisfying the following conditions: (a) the composite of F with the homomorphism End(P → C) → N ≥1 determined by considering the Frobenius degree is the iden- tity on N ≥1 ; (b) the endomorphisms of objects of C determined by an element of End(P → C) in the image of F are base-identity endomorphisms of Frobenius type. We shall refer to a Frobenius-section F which is regarded as being known only up to composition with automorphisms of the monoid N ≥1 as a quasi-Frobenius-section. If F is a Frobenius-section, then we shall refer to the endomorphisms of C induced by elements of End(P → C) in the image of F as F -distinguished. 52 SHINICHI MOCHIZUKI (iii) We shall refer to a pair (P, F ), where P is a base-section of C, and F is a P-Frobenius-section of C, as a base-Frobenius pair of C; when F is regarded as being known only up to composition with automorphisms of the monoid N ≥1 , we shall refer to such a pair as a quasi-base-Frobenius pair. If the Frobenioid C admits a base-Frobenius pair [or, equivalently, a quasi-base-Frobenius pair], then we shall say that C is of pre-model type. Remark 2.7.1. The notions of a “base-section” and “Frobenius-section” are intended to be a sort of “category-theoretic translation” of the notion of a “choice of trivialization of a trivial line bundle”, which occurs naturally when C is a category of trivial line bundles [cf. Remark 5.6.1; Examples 6.1, 6.3 below]. Remark 2.7.2. Suppose that C is of isotropic type. Let (P, F ) be a base-Frobenius pair of C. Then the only arrows of C which are both F - and P-distinguished [hence base-identity automorphisms — cf., e.g., the factorization of Deﬁnition 1.3, (iv), (a)] are the identity arrows. Suppose further that the Frobenioid C is of base-trivial type, and that the category C is a skeleton. Then every morphism φ of C admits a factorization φ = α ◦ β ◦ γ where α is P-distinguished; β is a base-identity pre-step endomorphism; γ is F - distinguished. Moreover, this factorization is unique [in the strict sense — i.e., not up to isomorphisms, etc.]. [Indeed, the existence and uniqueness of the factorization in question follow immediately from Deﬁnition 1.3, (iv), (a); the deﬁnition of P-, F -distinguished; our assumptions concerning C; the total epimorphicity of C; the isomorphism of functors appearing in the deﬁnition of a “pull-back morphism” in Deﬁnition 1.2, (ii).] Deﬁnition 2.8. (i) If, for every A ∈ Ob(C), it holds that O × (A) admits a [uniquely determined] proﬁnite topology such that O × (A), equipped with this topology, is a topologically ﬁnitely generated proﬁnite [abelian] group, then we shall say that C is of unit- proﬁnite type. (ii) Suppose that M is a topologically ﬁnitely generated proﬁnite abelian group. Thus, M decomposes as a direct product of pro-l groups M [l], where l varies over the elements of Primes [cf. §0]. We shall refer to the factor M [l] as the pro-l portion of M . (iii) Let M be as in (ii); assume that the group law of M is written multiplica- tively. If ζ : Primes → N ≥1 is a set-theoretic function, then we shall refer to as the map given by raising to the ζ-th power on M the map M → M (M ) a → a ζ (∈ M ) given by raising to the ζ(l)-th power on M [l], for l ∈ Primes. We shall refer to a set-theoretic function ζ : Primes → N ≥1 as being of co-prime type if ζ maps each THE GEOMETRY OF FROBENIOIDS I 53 element l ∈ Primes to an element of N ≥1 that is prime to l. [Thus, if ζ is of co-prime type, then the map given by raising to the ζ-th power will always be bijective.] Proposition 2.9. (Unit-wise Frobenius Functors II) Suppose that the Frobenioid C is of Frobenius-normalized, base-trivial, isotropic, and Aut- ample [cf. Remark 2.9.2 below] type. Then: (i) If the base category D admits a terminal object [cf. §0], then C is of pre-model type. (ii) Let τ be a characteristic splitting on C; ζ : Primes → N ≥1 a set-theoretic function. Suppose that C is of pre-model and unit-proﬁnite type. Then there exists a functor Ψ: C→C — which we shall refer to as the unit-wise Frobenius functor [associated to τ , ζ] — which satisﬁes the following properties: (a) Ψ is 1-compatible, relative to the functor C → F Φ , with the identity functor on F Φ . (b) Ψ maps an object (respectively, morphism of Frobenius type; pre-step; pull-back morphism) of C to an isomorphic object (respectively, abstractly equivalent morphism; abstractly equivalent morphism; abstractly equivalent morphism) of C. (c) If A ∈ Ob(C), then there exists an isomorphism Ψ(A) ∼ = A such that the endomorphism of O × (A) induced by Ψ followed by conjugation by this isomorphism is given by raising to the ζ-th power. (d) If ζ is of co-prime type [cf. Deﬁnition 2.8, (iii)], then Ψ is an equiv- alence of categories. If C is of unit-trivial type, then Ψ is isomorphic to the identity functor. Proof. By well-known general nonsense in category theory, we may assume, with- out loss of generality, for the remainder of the proof of Proposition 2.9, that the category C is a skeleton. Thus, C pl-bk is also a skeleton. Now we consider assertion (i). Observe [cf. Deﬁnition 1.3, (i), (c); the fact that C is of base-trivial type] that if A ∈ Ob(C) projects to a terminal object of D, then A is pseudo-terminal [cf. §0]. Note that by Deﬁnition 1.3, (i), (a), and our assumptions on D, it follows that such an object A always exists; let us ﬁx one such object A. Thus, the natural projection functor determines an equivalence of categories ∼ pl-bk C A →D [cf. Deﬁnition 1.3, (i), (c)]. Note that it follows immediately from the existence of this equivalence of categories [together with the fact that A maps to a terminal 54 SHINICHI MOCHIZUKI pl-bk object of D] that the natural functor C A → C pl-bk is injective on isomorphism pl-bk is a skeletal subcategory [cf. §0], then classes of objects. In particular, if Q ⊆ C A [relative to some suﬃciently large universe with respect to which, say, the category C is small] the natural map Ob(Q) → Ob(C pl-bk ) = Ob(C) ∼ pl-bk is bijective [cf. the fact that C is of base-trivial type; the equivalence C A → D]. Thus, the subcategory P ⊆ C determined by the image of the objects and arrows of Q in C is a skeleton which satisﬁes the conditions of Deﬁnition 2.7, (i) [cf. the fact that C is of base-trivial and isotropic type; Deﬁnition 1.3, (i), (a)] — that is to say, P is a base-section. Next, let us observe [cf. the fact that C is of base-trivial and isotropic type; Def- inition 1.3, (i), (a)] that A is Frobenius-trivial, hence that there exists a morphism of monoids F A : N ≥1 → End C (A) whose composite with the morphism of monoids deg Fr (−) : End C (A) → N ≥1 is the identity morphism on N ≥1 , and whose image consists of base-identity endomor- phisms of Frobenius type of A. Thus, by Proposition 1.11, (iii), we conclude that F A extends to a P-Frobenius-section F : N ≥1 → End(P → C) — hence that C is of pre-model type, as desired. This completes the proof of assertion (i). Next, we consider assertion (ii). Observe [cf. the fact that C is of base-trivial and isotropic type; the fact that C is a skeleton] we may apply Remark 2.7.2 to conclude that every morphism φ : C → D of C admits a unique factorization φ = α ◦ β ◦ γ in C, where α is P-distinguished; β is a base-identity pre-step endomorphism; γ is F -distinguished. Now [cf. the proof of Proposition 2.5, (iii)] by applying the ∼ characteristic splitting [cf. Deﬁnition 2.3, (a)] O × (C) × τ (C) → O (C), we may write β = β 0 · β 1 ∈ O (C) [where β 0 ∈ O × (C), β 1 ∈ τ (C)]. Set def Ψ(β) = β 0 ζ · β 1 ; def Ψ(φ) = α ◦ Ψ(β) ◦ γ [where “(−) ζ ” is as deﬁned in Deﬁnition 2.8, (iii)]. Since β is completely determined by φ, it follows that Ψ is well-deﬁned [as a “map on arrows”]. Moreover, it follows from the deﬁnition of P- and F -distinguished morphisms [together with the fact that raising to the ζ-th power deﬁnes an endomorphism of the functor in monoids “O × (−)” on C lin which commutes with raising to the d-th power, for d ∈ N ≥1 — cf. THE GEOMETRY OF FROBENIOIDS I 55 our assumption that C is of Frobenius-normalized type] that Ψ is, in fact, a functor, and that Ψ satisﬁes properties (a), (b), (c), (d) in the statement of Proposition 2.9, (ii). This completes the proof of Proposition 2.9. Remark 2.9.1. By “base-changing” the Frobenioid C via various functors D → D as in Proposition 1.6, it follows that one may obtain “unit-wise Frobenius functors” as in Proposition 2.9, (ii), for many Frobenioids whose base categories do not nec- essarily admit terminal objects [as is required in the hypotheses of Proposition 2.9, (i)]. Remark 2.9.2. We shall see later [cf. Theorem 5.1, (iii)] that in fact, the Aut-ampleness hypothesis in the statement of Proposition 2.9 is superﬂuous. 56 SHINICHI MOCHIZUKI Section 3: Category-theoreticity of the Base and Frobenius Degree In the present §3, we show various results in the “opposite direction” to the direction represented by the various Frobenius functors of §2. Namely, we show that various natural structures — such as Frobenius degrees and the natural projection functor to the base category — are preserved by equivalences of categories between Frobenioids. In the following discussion, we maintain the notation of §1, §2. Also, we assume that we have been given a divisorial monoid Φ on a connected, totally epimorphic category D and a Frobenioid C → F Φ . Deﬁnition 3.1. (i) We shall say that C is of quasi-isotropic type if it holds that A ∈ Ob(C) is non-isotropic if and only if it is an iso-subanchor [cf. §0]. [Thus, if C is of isotropic type, then C is of quasi-isotropic type — cf. Remark 3.1.1 below.] We shall say that C is of standard type if the following conditions are satisﬁed: (a) C is of quasi-isotropic and Frobenius-isotropic type; (b) if C is of group-like type, then C istr admits a Frobenius-compact object; (c) C is of Frobenius-normalized type; (d) D is of FSMFF-type; (e) Φ is non-dilating. We shall say that a category E is Frobenius-slim if every homomorphism of monoids F → Aut(E A → E) [where A ∈ Ob(E)] factors through the natural surjection F N ≥1 . [Thus, every slim category is Frobenius-slim.] (ii) Write C Fr-tp ⊆ C for the subcategory of C determined by the morphisms of Frobenius type; C bi-Fr ⊆ C Fr-tp × C Fr-tp for the subcategory of the product category C Fr-tp × C Fr-tp determined by pairs of morphisms of Frobenius type of the same Frobenius degree. For A, B ∈ Ob(C), we shall write def Hom pf C (A, B) = lim − → (A→A ,B→B )∈Ob( (A,B) C bi-Fr ) Hom C (A , B ) where the inductive limit is parametrized by [say, some small skeletal subcategory of] (A,B) C bi-Fr ; the map Hom C (A , B ) → Hom C (A , B ) induced by a morphism (A → A , B → B ) in (A,B) C bi-Fr from an object (A → A , B → B ) of (A,B) C bi-Fr to an object (A → A , B → B ) of (A,B) C bi-Fr is the map determined by the assignment “φ → φ ” of Proposition 1.10, (i). We shall refer to an element of Hom pf C (A, B) as a perfected morphism A → B. (iii) Suppose that the Frobenioid C is of Frobenius-isotropic type. Then we shall write C pf THE GEOMETRY OF FROBENIOIDS I 57 for the category — which we shall refer to as the perfection of C — deﬁned as follows: The objects of C pf are pairs (A, n), where A ∈ Ob(C), n ∈ N ≥1 . The morphisms of C pf are given by def Hom C pf ((A, n), (B, m)) = Hom pf C (A , B ) where (A, n) and (B, m) are objects of C pf ; A → A is a morphism of Frobenius type in C of Frobenius degree m; B → B is a morphism of Frobenius type in C of Frobenius degree n; one veriﬁes immediately [cf. Deﬁnition 1.3, (ii)] that this set of morphisms of C pf from (A, n) to (B, m) is independent [up to uniquely determined natural bijections] of the choice of morphisms of Frobenius type A → A , B → B ; composition of morphisms of C pf is deﬁned in the evident fashion. [Thus, in words, the pair (A, n) is to be thought of as an “n-th root of A”.] Also, we obtain a natural functor C → C pf [by mapping “A → (A, 1)”]. (iv) Two co-objective [cf. §0] morphisms α 1 , α 2 : A → B of C istr will be called unit-equivalent if there exist morphisms γ : A → C, β : C → B [in C istr ] and an automorphism δ ∈ O × (C) such that α 1 = β ◦ γ, α 2 = β ◦ δ ◦ γ. In this situation, O × we shall write α 1 ≈ α 2 . [Thus, if C is of unit-trivial type, then two co-objective morphisms of C istr are unit-equivalent if and only if they are equal.] By Proposition O × 3.3, (ii), below, it follows that “ ≈ ” determines an equivalence relation on the set of morphisms A → B in C istr which is, moreover, closed under composition of morphisms; we shall write Hom un-tr C istr (A, B) for the set of unit-equivalence classes of morphisms A → B. Also, we shall write C un-tr for the category whose objects are the objects of C istr , and whose morphisms are un-tr as the unit-trivialization of C. given by “Hom un-tr C istr (−, −)”, and refer to C Remark 3.1.1. Observe that: An iso-subanchor of the Frobenioid C is never isotropic. [In particular, if C is of isotropic type, then C is of quasi-isotropic type.] Indeed, by Proposition 1.10, (iv), an anchor is never isotropic. Thus, by Deﬁnition 1.3, (vii), (b), a subanchor is never isotropic. Now if B → A is a mono-minimal categorical quotient [cf. §0] in C of B by a group G ⊆ Aut C (B) such that B is a subanchor and A is isotropic, then applying the isotropiﬁcation functor of Proposition 1.9, (v), yields a factorization B → B → A, where B → B is an isotropic hull [hence a monomorphism — cf. Deﬁnition 1.3, (v), (a)], such that G acts compatibly [relative to the arrow B → B ] on B ; thus, by the deﬁnition of the term “mono-minimal” it follows that the arrow B → B is an isomorphism, i.e., that B is isotropic — a contradiction. This completes the proof of the “observation”. 58 SHINICHI MOCHIZUKI Remark 3.1.2. Observe that for any residually ﬁnite group G [i.e., a group G such that the intersection of the normal subgroups of ﬁnite index of G is trivial]: Any homomorphism of monoids F → G factors through the natural sur- jection F N ≥1 . [Indeed, it suﬃces to show this when G is ﬁnite. When G is ﬁnite, it follows immediately from the deﬁnition of F [cf. Deﬁnition 1.1, (iii)] that the image of 1 ∈ Z ≥0 in G is an element γ ∈ G such that for every d ∈ N ≥1 , there exists an element δ d ∈ G such that δ d ·γ ·δ d −1 = γ d . Thus, by taking d to be the order of γ, we conclude that γ is the identity, hence that the homomorphism of monoids F → G factors through the natural surjection F N ≥1 , as desired.] In particular, it follows that if E is a category such that for every A ∈ Ob(E), the group Aut(E A → E) is residually ﬁnite, then E is Frobenius-slim. Remark 3.1.3. The phenomenon discussed in Remark 3.1.2 may be regarded as an example of the following fundamental dichotomy [which is, in a certain sense, a central theme of the theory of the present paper] between the structure of the base category D and the “Frobenius structure” constituted by N ≥1 : base category “indiﬀerent to order” groups Frobenius “order-conscious” non-group-like monoids This sort of phenomenon may be observed in “classical scheme theory” for instance in the invariance of the étale site of a scheme in positive characteristic under the Frobenius morphism. Here, it is useful to recall that a typical example of a “base category” is constituted by the subcategory of connected objects of a Galois category [which is easily veriﬁed to be of FSM-, hence also of FSMFF-type]. By contrast, categories such as Order(Z ≥0 ), Order(N ≥1 ) or [the one-object categories determined by] Z ≥0 , N ≥1 are not of FSMFF-type. In this context, it is interesting to note that categories such as Order(−) of a ﬁnite subset of Z ≥0 of cardinality ≥ 2 [with the induced ordering] constitute a sort of “borderline case”, which is of FSMFF-, but not of FSM-, type. Proposition 3.2. (Perfections of Frobenioids) Suppose that the Frobenioid C is of Frobenius-isotropic type. Then: (i) There is a natural 1-commutative diagram of functors C ⏐ ⏐ −→ C pf ⏐ ⏐ F Φ −→ F Φ pf — where the vertical arrow on the left is the functor that deﬁnes the Frobenioid structure on C; the vertical arrow on the right is induced by the vertical arrow on THE GEOMETRY OF FROBENIOIDS I 59 the left; the lower horizontal arrow is induced by the natural morphism of monoids Φ → Φ pf ; the upper horizontal arrow is the natural functor C → C pf of Deﬁnition 3.1, (iii). In particular, the functor C pf → F Φ pf determines a pre-Frobenioid structure on C pf . (ii) An arrow of C pf is a(n) morphism of Frobenius type (respectively, pre-step; base-isomorphism; base-identity endomorphism; isomorphism; pull-back morphism; isometry; co-angular morphism; LB-invertible mor- phism; morphism of a given Frobenius degree) if and only if a coﬁnal collection of the system of arrows of C that determine this arrow of C pf [cf. Deﬁnition 3.1, (ii)] is so. (iii) The category C pf , equipped with the functor C pf → F Φ pf of the diagram of (i), is a Frobenioid of perfect and isotropic type. Moreover, there is a natural ∼ equivalence of categories C pf → (C pf ) pf . Proof. In light of our assumption that the Frobenioid C is of Frobenius-isotropic type, assertions (i), (ii), (iii) follow immediately from the deﬁnitions; Proposition 1.10, (i) [cf. also Proposition 2.1, (iii)]. Proposition 3.3. (Base-identity Pre-steps and Units) (i) Write pl-bk pl-bk End(C A → C) bs-iso ⊆ End(C A → C) pl-bk [where C pl-bk is as in Deﬁnition 1.3, (i), (c); C A → C is the natural functor] for the submonoid consisting of those natural transformations such that the various endomorphisms of objects of C that occur in the natural transformation are all base- isomorphisms. Then if D is Frobenius-slim, then the image of 1 ∈ Z ≥0 ⊆ F under any homomorphism of monoids pl-bk → C) bs-iso F → End(C A pl-bk determines an element of End(C A → C) bs-iso with the property that the various endomorphisms of objects of C that occur in the natural transformation determined by this element are all base-identity pre-steps [i.e., lie in “O (−)”]. Conversely, if C is of Frobenius-normalized type, and A is Frobenius-trivial, then every base-identity pre-step endomorphism of A arises as the endomorphism of A induced pl-bk by the image of 1 ∈ Z ≥0 ⊆ F via a homomorphism of monoids F → End(C A → bs-iso . C) (ii) Two co-objective morphisms α 1 , α 2 : A → B of C istr are unit-equivalent if and only if they map to the same morphism of F Φ , i.e., if and only if the following three conditions are satisﬁed: (a) deg Fr (α 1 ) = deg Fr (α 2 ); (b) Div(α 1 ) = Div(α 2 ); (c) Base(α 1 ) = Base(α 2 ). (iii) There is a natural functor C istr → C un-tr 60 SHINICHI MOCHIZUKI which is full and essentially surjective; moreover, this functor is an equivalence of categories if and only if C istr is of unit-trivial type. (iv) The functor C istr → F Φ factors naturally through C un-tr , hence determines a functor C un-tr → F Φ which is faithful and essentially surjective; moreover, this functor determines a natural structure of Frobenioid on C un-tr , with respect to which C un-tr is of isotropic and unit-trivial type. Finally, an arrow of C un-tr is a(n) morphism of Frobenius type (respectively, pre-step; base-isomorphism; isomorphism; pull-back morphism; isometry; co-angular morphism; LB-invertible mor- phism; morphism of a given Frobenius degree) if and only if it arises from such an arrow of C istr . (v) The functor C → F Φ is an equivalence of categories if and only if the Frobenioid C is of Aut-ample, unit-trivial, and base-trivial type. Proof. First, we consider assertion (i). Note that since the composite of the functor pl-bk C A → C with the natural projection functor C → D factors as the composite of the equivalence of categories [involving pull-back morphisms] of Deﬁnition 1.3, (i), ∼ def pl-bk (c), C A → D A D [where A D = Base(A)] with the natural functor D A D → D, it pl-bk → C) bs-iso determines follows that any homomorphism of monoids F → End(C A a homomorphism of monoids F → Aut(D A D → D) — which, if D is Frobenius-slim [cf. Deﬁnition 3.1, (i)], necessarily factors through the natural surjection F N ≥1 — together with a homomorphism of monoids F → N ≥1 obtained by considering the Frobenius degree of the induced endomorphism of A — which [in light of the fact that the monoid N ≥1 is commutative, together with the structure of F — cf. Deﬁnition 1.1, (iii)] also necessarily factors through the natural surjection F N ≥1 . Thus, we conclude that if D is Frobenius-slim, then the image pl-bk → of 1 ∈ Z ≥0 ⊆ F under the given homomorphism of monoids F → End(C A pl-bk bs-iso bs-iso determines an element of End(C A → C) with the property that the C) various endomorphisms of objects of C that occur in the natural transformation determined by this element are all base-identity pre-steps, as desired. The “converse assertion” [when C is of Frobenius-normalized type, and A is Frobenius-trivial] may be veriﬁed by choosing a homomorphism of monoids N ≥1 → End C (A) THE GEOMETRY OF FROBENIOIDS I 61 as in the deﬁnition of the term “Frobenius-trivial” [cf. the homomorphism “ζ” of Deﬁnition 1.2, (iv)], which, together with the homomorphism of monoids Z ≥0 → End C (A) that maps 1 ∈ Z ≥0 to a given base-identity pre-step endomorphism of A, yields [cf. our assumption that C is of Frobenius-normalized type!] a homomorphism of monoids F → End C (A) — which, by applying Proposition 1.11, (iii), lifts to a homomorphism of monoids pl-bk → C) bs-iso , as desired. This completes the proof of assertion (i). F → End(C A Next, we consider assertion (ii). Since assertion (ii) clearly only concerns the Frobenioid C istr [cf. Proposition 1.9, (v)], we may replace C by C istr and assume for the remainder of the proof of assertion (ii) that C is of isotropic type. Now the necessity of the three conditions (a), (b), (c) follows immediately [cf. Remark 1.1.1] from the fact that endomorphisms of “O × ” are LB-invertible base-identity linear endomorphisms. To show the suﬃciency of these three conditions, we apply the factorization of Deﬁnition 1.3, (iv), (a) [cf. also Proposition 1.4, (i)], the essential uniqueness of morphisms of Frobenius type of a given Frobenius degree [cf. Deﬁ- nition 1.3, (ii)], and the equivalence of categories [involving pull-back morphisms] of Deﬁnition 1.3, (i), (c), to α 1 , α 2 . Then conditions (a), (c) imply that there exist morphisms γ : A → C; β 1 , β 2 : C → D; δ : D → B, where γ is a mor- phism of Frobenius type, β 1 and β 2 are base-equivalent co-angular pre-steps, and δ : D → B is a pull-back morphism such that α 1 = δ ◦ β 1 ◦ γ, α 2 = δ ◦ β 2 ◦ γ. Since δ, γ are LB-invertible, it thus follows from condition (b) [cf. also Remark 1.1.1] that Div(β 1 ) = Div(β 2 ), hence [by Deﬁnition 1.3, (vi)] that β 2 = ζ ◦ β 1 , for some ζ ∈ O × (D). Since α 1 = δ ◦ (β 1 ◦ γ), α 2 = δ ◦ ζ ◦ (β 1 ◦ γ), we thus conclude that O × α 1 ≈ α 2 , as desired. This completes the proof of assertion (ii). Now assertion (iii) is immediate from the deﬁnitions. In light of assertions (ii), (iii), assertion (iv) is immediate from the deﬁnitions. As for assertion (v), the necessity of the condition in the statement of assertion (v) follows immediately from Proposition 1.5, (i), (ii). To verify the suﬃciency of this condition, let us ﬁrst observe that if C is of unit-trivial and base-trivial type, then [by the existence of isotropic hulls in C — cf. Deﬁnition 1.3, (vii), (a)] it follows that C is also of isotropic type, hence that we have a natural equivalence of categories ∼ C → C un-tr [cf. assertion (iii)]. Thus, by assertion (iv), it follows that the natural functor C → F Φ is faithful and essentially surjective. Since C is of base-trivial and Aut-ample type, it follows from the factorization of Deﬁnition 1.3, (iv), (a) [cf. also the existence and uniqueness of morphisms of Frobenius type of a given Frobenius degree asserted in Deﬁnition 1.3, (ii); the equivalence of categories involving pull- back morphisms of Deﬁnition 1.3, (i), (c)], that to show that C → F Φ is full, it suﬃces to show that it is surjective on base-identity pre-step endomorphisms [i.e., on “O (−)”]; but, by our assumption that C is of base-trivial and Aut-ample type, this follows immediately from the ﬁrst equivalence of categories of Deﬁnition 1.3, (iii), (d). This completes the proof of assertion (v). 62 SHINICHI MOCHIZUKI Theorem 3.4. (Category-theoreticity of the Base and Frobenius De- gree) For i = 1, 2, let Φ i be a divisorial monoid on a connected, totally epimor- phic category D i ; C i → F Φ i a Frobenioid; ∼ Ψ : C 1 → C 2 an equivalence of categories. Then: (i) Suppose that C 1 , C 2 are of quasi-isotropic type. Then Ψ preserves the isotropic objects, isotropic hulls, and isometric pre-steps. Moreover, there exists a 1-unique functor Ψ istr : C 1 istr → C 2 istr that ﬁts into a 1-commutative diagram C 1 ⏐ ⏐ C 1 istr Ψ −→ C 2 ⏐ ⏐ Ψ istr −→ C 2 istr [where the vertical arrows are the natural “isotropiﬁcation functors” of Proposition 1.9, (v); the horizontal arrows are equivalences of categories]. Finally, if D 1 , D 2 are slim, and C 1 , C 2 are of Frobenius-normalized type, then each of the composite functors of this diagram is rigid. (ii) Suppose that C 1 , C 2 are of quasi-isotropic type, and that D 1 , D 2 are of FSMFF-type. Then Ψ preserves pre-steps, co-angular pre-steps, and group- like objects. (iii) Suppose that: (a) C 1 , C 2 are of standard type; (b) if C 1 , C 2 are of group- like type, then both Ψ and some quasi-inverse to Ψ preserve base-isomorphisms. Then Ψ preserves morphisms of Frobenius type, linear morphisms, base- isomorphisms, co-angular morphisms, pull-back morphisms, isometries, and LB-invertible morphisms. Moreover, there exists an automorphism of monoids ∼ Ψ N ≥1 : N ≥1 → N ≥1 such that Ψ maps morphisms of Frobenius degree d to morphisms of Frobenius degree Ψ N ≥1 (d); if C 1 , C 2 admit a non-group-like object, then Ψ N ≥1 is the identity pf pf automorphism. Also, there exists a 1-unique functor Ψ : C 1 → C 2 pf that ﬁts into a 1-commutative diagram Ψ C 1 −→ C 2 ⏐ ⏐ ⏐ ⏐ C 1 pf Ψ pf −→ C 2 pf [where the vertical arrows are the natural functors of Proposition 3.2, (i); the hori- zontal arrows are equivalences of categories]. Finally, if D 1 , D 2 are slim, then each of the composite functors of this diagram is rigid. (iv) Suppose that: (a) C 1 , C 2 are of standard type; (b) if C 1 , C 2 are of group- like type, then both Ψ and some quasi-inverse to Ψ preserve base-isomorphisms; THE GEOMETRY OF FROBENIOIDS I 63 (c) D 1 , D 2 are Frobenius-slim. Then Ψ preserves the submonoids “O (−)”, “O × (−)”; Ψ N ≥1 is the identity automorphism. Moreover, there exists a 1-unique functor Ψ un-tr : C 1 un-tr → C 2 un-tr that ﬁts into a 1-commutative diagram C 1 istr ⏐ ⏐ C 1 un-tr Ψ istr −→ Ψ un-tr −→ C 2 istr ⏐ ⏐ C 2 un-tr [where the vertical arrows are the natural functors of Proposition 3.3, (iii); the horizontal arrows are equivalences of categories]. Finally, if D 1 , D 2 are slim, then each of the composite functors of this diagram is rigid. (v) Suppose that: (a) C 1 , C 2 are of standard type; (b) if C 1 , C 2 are of group- like type, then both Ψ and some quasi-inverse to Ψ preserve base-isomorphisms; (c) D 1 , D 2 are slim. Then Ψ preserves the base-identity endomorphisms and base-equivalent pairs of co-objective morphisms. Moreover, there exists a 1- unique functor Ψ Base : D 1 → D 2 that ﬁts into a 1-commutative diagram C 1 ⏐ ⏐ D 1 Ψ −→ Ψ Base −→ C 2 ⏐ ⏐ D 2 [where the vertical arrows are the natural projection functors; the horizontal ar- rows are equivalences of categories]. Finally, each of the composite functors of this diagram is rigid. Proof. First, we consider assertion (i). Since iso-subanchors are manifestly pre- served by any equivalence of categories, it follows from our assumption that C 1 , C 2 are of quasi-isotropic type that Ψ preserves isotropic objects. Now, with the excep- tion of the ﬁnal statement concerning the rigidity of the composite functors, the remainder of assertion (i) follows formally from [the deﬁnitions and] Proposition 1.9, (v), (vi), (vii). The ﬁnal statement concerning the rigidity of the composite functors may be veriﬁed as follows: By Proposition 1.13, (ii), it suﬃces to show, for each A ∈ Ob(C istr ) that the automorphism α ∈ O × (A) induced by an automor- phism ∈ Aut(C 1 → C 1 istr ) is trivial. But, by Deﬁnition 1.3, (i), (a), (b); (iii), (c), it suﬃces to show this when A is Frobenius-trivial, in which case the triviality of α follows from the functoriality of α with respect to base-identity endomorphisms of A of arbitrary of Frobenius degree [which implies, since C 1 , C 2 are of Frobenius- normalized type, that α d = α, for all d ∈ N ≥1 , hence that α is trivial, as desired]. Next, we consider assertion (ii). By assertion (i) [cf. also Proposition 1.9, (v)], and the characterization of co-angular pre-steps given in Proposition 1.7, (iv), we reduce immediately to the case where C 1 , C 2 are of isotropic type. Then [since any equivalence of categories manifestly preserves FSM-morphisms and irreducible mor- phisms] the fact that Ψ preserves pre-steps follows formally from Proposition 1.14, (ii), (iii). Since Ψ preserves pre-steps, it thus follows from Proposition 1.8, (iii) [cf. 64 SHINICHI MOCHIZUKI also Proposition 1.4, (i)], that Ψ preserves group-like objects. This completes the proof of assertions (i), (ii). Next, we consider assertion (iii). First, I claim that to verify assertion (iii), it suﬃces to prove that, for each prime p 1 ∈ Primes, there exists a prime p 2 ∈ Primes, which is equal to p 1 if C 1 , C 2 are not of group-like type, such that Ψ istr maps p 1 - Frobenius morphisms to p 2 -Frobenius morphisms. Indeed, the assignment p 1 → p 2 determines a homomorphism of monoids Ψ N ≥1 : N ≥1 → N ≥1 which [by considering a quasi-inverse to Ψ] is easily seen to be an automorphism. Moreover, by Proposition 1.10, (v), the condition of the claim implies that Ψ istr preserves morphisms of Frobenius type, hence also linear morphisms [by Proposition 1.7, (iii)], and maps morphisms of Frobenius degree d to morphisms of Frobenius degree Ψ N ≥1 (d) [i.e., since arbitrary morphisms may be written as composites of prime-Frobenius morphisms and linear morphisms — cf. Remark 1.1.1; Deﬁnition 1.3, (iv), (a); Proposition 1.10, (v)]. Since the isotropiﬁcation functor preserves Frobenius degrees, this implies that Ψ maps morphisms of Frobenius degree d to morphisms of Frobenius degree Ψ N ≥1 (d), hence that Ψ preserves linear morphisms and morphisms of Frobenius type [by Proposition 1.7, (iii)]. Moreover, by assertions (i), (ii), Ψ preserves isometric pre-steps and pre-steps, hence base-isomorphisms [i.e., composites of pre-steps and morphisms of Frobenius type — cf. Proposition 1.7, (ii)], pull-back morphisms [cf. Proposition 1.7, (ii)], isometries [i.e., morphisms that map via the isotropiﬁcation functor to composites of a morphism of Frobenius type and a pull-back morphism — cf. Propositions 1.4, (i), (v); 1.9, (v)], co-angular morphisms [cf. Deﬁnition 1.2, (iii); assertion (i) for isometric pre-steps], and LB- invertible morphisms. Now it follows immediately from the deﬁnition of C pf [cf. Deﬁnition 3.1, (iii)] that we obtain a 1-unique 1-commutative diagram as in the statement of assertion (iii). Finally, to verify the asserted rigidity of composite functors, it suﬃces [cf. the argument applied in the proof of assertion (i)] to apply Proposition 1.13, (ii), and to consider the functoriality of the automorphisms in question with respect to base-identity endomorphisms of Frobenius-trivial objects of arbitrary Frobenius degree. This completes the proof of the claim. Thus, to complete the proof of assertion (iii), we may assume [for the remainder of the proof of assertion (iii)] that C 1 , C 2 are of isotropic type [cf. assertion (i)]. Then it suﬃces to prove that, for each prime p 1 ∈ Primes, there exists a prime p 2 ∈ Primes, which is equal to p 1 if C 1 , C 2 are not of group-like type, such that Ψ maps p 1 - Frobenius morphisms to p 2 -Frobenius morphisms. Let us call an object A 1 ∈ Ob(C 1 ) (p 1 , p 2 )-admissible if Ψ maps every p 1 -Frobenius morphism with domain A 1 to a p 2 -Frobenius morphism of C 2 . Now let us consider the following assertions: (F1) For each prime p 1 ∈ Primes, there exist a prime p 2 ∈ Primes and a (p 1 , p 2 )-admissible object of C 1 . (F2) For every pair of primes p 1 , p 2 ∈ Primes and every morphism ζ 1 : A 1 → B 1 in C 1 , A 1 is (p 1 , p 2 )-admissible if and only if B 1 is. THE GEOMETRY OF FROBENIOIDS I 65 (F3) If C 1 , C 2 are not of group-like type, then for each prime p ∈ Primes, there exist a (p, p)-admissible object of C 1 . Observe, moreover, that since C 1 is connected, to complete the proof of assertion (iii), it suﬃces to prove (F1), (F2), (F3). First, we consider assertion (F1). Let us ﬁrst consider the case where C 1 , C 2 are of group-like type. Then all pre-steps of C 1 , C 2 are isomorphisms; Ψ preserves base-isomorphisms. Thus, for any A 1 ∈ Ob(C 1 ), the prime-Frobenius morphisms with domain A 1 are precisely the irreducible base-isomorphisms with domain A 1 [cf. Proposition 1.14, (i)]. In particular, Ψ preserves the prime-Frobenius morphisms; hence, we conclude that assertion (F1) holds. Next, let us consider the case where C 1 , C 2 are not of group-like type. Then if A 1 is non-group-like, then [cf. Deﬁnition 1.3, (i), (a); Proposition 1.8, (iii)], there exists a base-isomorphic [i.e., to A 1 ], hence non-group-like, Frobenius-trivial object of C 1 . Thus, we may assume without loss of generality that A 1 is Frobenius-trivial. Then for any p 1 ∈ Primes, there exists a base-identity [hence Div-identity] p 1 -Frobenius endomorphism φ 1 of A 1 . Since [by assertion (ii)] Ψ preserves pre-steps, it thus follows formally from the characteriza- tion of “Div-identity prime-Frobenius endomorphisms” given in Proposition 1.14, def (v), that Ψ maps φ 1 to a prime-Frobenius endomorphism of A 2 = Ψ(A 1 ). This completes the proof of assertion (F1). Next, we consider assertion (F2). First, observe that if the morphism ζ 1 : A 1 → B 1 is a pre-step, then [since, by assertion (ii), Ψ preserves pre-steps] it follows by applying Proposition 1.14, (iv), to commutative diagrams such as the one given in Proposition 1.10, (i), that assertion (F2) holds. Thus, by Deﬁnition 1.3, (i), (a), (b), (c), we may assume without loss of generality that B 1 is Frobenius-trivial, and that ζ 1 is a pull-back morphism. Now, by applying Proposition 1.11, (iii), it follows that for every p 1 ∈ Primes, there exist base-identity p 1 -Frobenius endomorphisms φ 1 ∈ End C (A 1 ), ψ 1 ∈ End C (B 1 ) such that ψ 1 ◦ ζ 1 = ζ 1 ◦ φ 1 . In particular, if we def def def write φ 2 = Ψ(φ 1 ), ψ 2 = Ψ(ψ 1 ), ζ 2 = Ψ(ζ 1 ), then ψ 2 ◦ ζ 2 = ζ 2 ◦ φ 2 , and φ 2 , ψ 2 are irreducible. Thus, by Proposition 1.14, (iv), we obtain that φ 2 is a p 2 -Frobenius morphism if and only if ψ 2 is. This completes the proof of assertion (F2). Finally, we consider assertion (F3). Let A 1 ∈ Ob(C 1 ) be a non-group-like, Frobenius-trivial object [cf. the proof of assertion (F1)]. By assertions (F1), (F2), it follows already that Ψ preserves prime-Frobenius morphisms. Thus, to complete the proof of assertion (F3), [since the Frobenius degree of a prime-Frobenius mor- phism is always a prime number] it suﬃces to show that if ζ 1 , θ 1 ∈ End C 1 (A 1 ) are prime-Frobenius base-identity endomorphisms such that deg Fr (ζ 1 ) < deg Fr (θ 1 ), def def then deg Fr (ζ 2 ) < deg Fr (θ 2 ) [where ζ 2 = Ψ(ζ 1 ), θ 2 = Ψ(θ 1 )]. But, by the ﬁrst equivalence of categories of Deﬁnition 1.3, (iii), (d) [cf. also Proposition 1.10, (i)], the condition “deg Fr (ζ 1 ) < deg Fr (θ 1 )” is equivalent to the following condition: If we write β ζ 1 (respectively, β θ 1 ) for the step “β ◦ α” of Proposition 1.14, (v), obtained when one takes “φ” of loc. cit. to be ζ 1 (respectively, θ 1 ) [and “α” of loc. cit. to be some ﬁxed step], then β θ 1 = γ 1 ◦ β ζ 1 , for some step γ 1 . 66 SHINICHI MOCHIZUKI Thus, if we write β ζ 2 (respectively, β θ 2 ) for the step “β ◦ α” of Proposition 1.14, (v), obtained when one takes “φ” of loc. cit. to be ζ 2 (respectively, θ 2 ) [and “α” of loc. cit. to be some ﬁxed step], then β θ 2 = γ 2 ◦ β ζ 2 , for some step γ 2 [since, by assertion (ii), we already know that Ψ preserves pre-steps], which [again by the ﬁrst equivalence of categories of Deﬁnition 1.3, (iii), (d); Proposition 1.10, (i)] implies that deg Fr (ζ 2 ) < deg Fr (θ 2 ), as desired. This completes the proof of assertion (F3), hence also the proof of assertion (iii). Next, let us observe that by assertion (i) [cf. also Proposition 1.9, (v)], it suﬃces to verify assertions (iv), (v), under the further assumption that C 1 , C 2 are of isotropic type; thus, we assume for the remainder of the proof of Theorem 3.4 that C 1 , C 2 are of isotropic type. Also, to simplify notation [for the remainder of the proof of Theorem 3.4], let us write def P i = C i pl-bk [cf. Deﬁnition 1.3, (i), (c)], for i = 1, 2. Next, let us consider assertion (iv). Now, for i = 1, 2, it follows formally [in light of our assumption that D i is Frobenius-slim] from Proposition 3.3, (i) [cf. also Deﬁnition 1.3, (i), (a), (b); (iii), (c)], that if C ∈ Ob(C i ), then the endomorphisms of O (C) are precisely the endomorphisms γ ∈ End C i (C) such that the following condition is satisﬁed: There exist pre-steps φ : A → B, ψ : A → C and endomorphisms α ∈ End C i (A), β ∈ End C i (B) such that β ◦ φ = φ ◦ α, γ ◦ ψ = ψ ◦ α, and, moreover, α arises as the endomorphism of A induced by the image of 1 ∈ Z ≥0 ⊆ F via a homomorphism of monoids F → End((P i ) A → C i ) bs-iso . By assertions (ii), (iii), it follows that Ψ preserves pre-steps, base-isomorphisms, and pull-back morphisms, hence that Ψ preserves endomorphisms satisfying the above condition. Thus, we conclude that Ψ preserves the submonoids “O (−)”, “O × (−)”, as desired. The existence of a a 1-unique functor Ψ un-tr : C 1 un-tr → C 2 un-tr that ﬁts into a 1-commutative diagram as in the statement of assertion (iv) then follows formally from the deﬁnition of “C 1 un-tr ”, “C 2 un-tr ”; since “C 1 un-tr ”, “C 2 un-tr ” are of unit-trivial type, the asserted rigidity follows formally from Proposition 1.13, (ii). Thus, to complete the proof of assertion (iv), it suﬃces to show that Ψ N ≥1 is the identity automorphism. If C 1 , C 2 are not of group-like type, then this already follows formally from assertion (iii). Thus, let us assume for the remainder of the proof of assertion (iv) that C 1 , C 2 are of group-like type. Observe that there exists def an object A 1 ∈ Ob(C 1 ) such that A 2 = Ψ(A 1 ) is Frobenius-compact [cf. Deﬁnition 3.1; the fact that Ψ is an equivalence of categories]. By Proposition 1.10, (vi), A 1 , A 2 are Frobenius-trivial. Let φ 1 ∈ End C 1 (A 1 ) be a base-identity prime-Frobenius def endomorphism. By assertion (iii), φ 2 = Ψ(φ 1 ) is also a prime-Frobenius morphism. Write φ 2 = α ◦ ψ 2 , where ψ 2 is a base-identity prime-Frobenius endomorphism of THE GEOMETRY OF FROBENIOIDS I 67 A 2 , and α ∈ Aut C 2 (A 2 ) [cf. Deﬁnition 1.3, (ii)]. Now since C 1 , C 2 are of Frobenius- normalized type [cf. Deﬁnition 3.1, (i), (c)], it follows that for every u 1 ∈ O × (A 1 ), def u p 1 1 ◦ φ 1 = φ 1 ◦ u 1 [where p 1 = deg Fr (φ 1 )]. Thus, for u 2 ∈ O × (A 2 ), we obtain u p 2 1 ◦ φ 2 = φ 2 ◦ u 2 = α ◦ ψ 2 ◦ u 2 = α ◦ u p 2 2 ◦ ψ 2 = α ◦ u p 2 2 ◦ α −1 ◦ α ◦ ψ 2 = α ◦ u p 2 2 ◦ α −1 ◦ φ 2 def [where p 2 = deg Fr (φ 2 )], hence [by the total epimorphicity of C 2 ] u p 2 1 = α ◦ u p 2 2 ◦ α −1 — i.e., α acts on O × (A 2 ) pf by multiplication by p 1 /p 2 . Since A 2 is Frobenius- compact, we thus conclude that p 1 = p 2 . This completes the proof of assertion (iv). Finally, we consider assertion (v). Now, for i = 1, 2, if A ∈ Ob(C i ) = Ob(P i ), def A D = Base(A) ∈ Ob(D i ), then the natural projection functor C i → D i determines a natural equivalence of categories ∼ (P i ) A → (D i ) A D def [cf. Deﬁnition 1.3, (i), (c)]. Moreover, if A ∈ Ob(C i ) = Ob(P i ), A D = Base(A ) ∈ Ob(D i ), then any arrow A → A determines a functor (P i ) A → (P i ) A by sending an object φ : C → A of (P i ) A to the object C → A of (P i ) A which is the pull-back morphism of C i that appears in the factorization of the composite C → A → A given in Deﬁnition 1.3, (iv), (a). Moreover, one veriﬁes immediately that this functor ﬁts into a natural 1-commutative diagram (P i ) A ⏐ ⏐ −→ (D i ) A D −→ (D i ) A D (P i ) A ⏐ ⏐ [where the upper horizontal arrow is the functor just deﬁned; the vertical arrows are the equivalences that arise from the natural projection functor C i → D i ; the lower horizontal arrow is the natural functor [cf. §0] determined by the arrow A D → A D obtained by projecting the given arrow A → A to D i ]. Next, observe that since the category D i , hence also the categories (D i ) A D , (P i ) A , are slim, it follows that the collection of categories “(P i ) A ” [where i is ﬁxed; A ranges over the objects of C i ] and functors “(P i ) A → (P i ) A ” [arising from arrows A → A of C i ] determine a 2-slim [cf. Deﬁnition A.1, (i)] 2-category of 1-categories, whose “coarsiﬁcation” [cf. Deﬁnition A.1, (ii)] we denote by Q i , together with a natural functor C i → Q i 68 SHINICHI MOCHIZUKI [i.e., which maps A → (P i ) A , (A → A) → {(P i ) A → (P i ) A }]. Similarly, the collection of categories “(D i ) A D ” [where i is ﬁxed; A D ranges over the objects of D i ] and functors “(D i ) A D → (D i ) A D ” [arising from arrows A D → A D of D i ] determine a 2-category of 1-categories, whose coarsiﬁcation we denote by E i , together with a natural functor D i → E i — which, in fact, may be identiﬁed with the “slim exponentiation functor” of Propo- sition A.2, hence, in particular, is an equivalence of categories. Thus, since the nat- ural projection functor C i → D i is essentially surjective [cf. Deﬁnition 1.3, (i), (a)], it follows that the natural projection functor C i → D i induces a faithful, essentially surjective functor Q i → E i ∼ which may be composed with a quasi-inverse to the natural equivalence D i → E i just discussed to obtain a faithful, essentially surjective functor Q i → D i [which is well-deﬁned up to isomorphism]. def def Next, let us observe that if A, A ∈ Ob(C i ), A D = Base(A), A D = Base(A ), then any morphism φ D : A D → A D may be written in the form φ D = Base(ψ) ◦ Base(γ) ◦ Base(α) −1 — where α : B → A, γ : B → C, are pre-steps; ψ : C → A is a pull-back morphism [cf. Deﬁnition 1.3, (i), (b), (c)]. Since [by the above discussion] any base- ∼ isomorphism ζ : D → E of C i induces an equivalence of categories (P i ) D D → (P i ) E D def def [where D, E ∈ Ob(C i ), D D = Base(D), E D = Base(E)], it thus follows that any collection of morphisms α, γ, ψ as just described determine a “new functor” (P i ) A D → (P i ) A D [i.e., by inverting the equivalence of categories induced by α and then composing with the functors induced by γ, ψ]. Thus, by enlarging the 2-slim 2-category of 1-categories considered above [i.e., whose coarsiﬁcation we called Q i ] by considering these “new functors”, we obtain a [slightly larger] 2-slim 2-category of 1-categories, whose coarsiﬁcation we denote by R i . In particular, we obtain a [faithful] embed- ding Q i → R i with the property that the functor Q i → D i considered above admits a natural extension to a functor R i → D i which [by the above discussion] is clearly an equivalence of categories. On the other hand, since [by assertions (ii), (iii)] Ψ preserves pre-steps, pull- back morphisms, and factorizations as in Deﬁnition 1.3, (iv), (a), it follows that Ψ THE GEOMETRY OF FROBENIOIDS I 69 induces a 1-commutative diagram Ψ C 1 ⏐ ⏐ −→ Q 1 ⏐ ⏐ −→ R 1 −→ R 2 Ψ Q C 2 ⏐ ⏐ Q 2 ⏐ ⏐ Ψ R — where the vertical functors are the natural functors of the above discussion, and the the horizontal functors are equivalences of categories induced by Ψ. Thus, ∼ by composing with the natural equivalences of categories R i → D i of the above discussion, we obtain a 1-commutative diagram as in the statement of assertion (v), which is clearly 1-unique [cf. Deﬁnition 1.3, (i), (a), (b), (c)]. Finally, the asserted rigidity follows formally from Proposition 1.13, (i). This completes the proof of assertion (v). Remark 3.4.1. With regard to assumption (b) of Theorem 3.4, (iii), (iv), (v), we observe the following: Suppose, in the situation of Theorem 3.4, that C 1 , C 2 are of group-like and quasi-isotropic type. Then if Ψ and some quasi-inverse to Ψ preserve Frobenius degrees, then they also preserve base-isomorphisms. Indeed, by Theorem 3.4, (i), we may assume, without loss of generality, that C 1 , C 2 are of isotropic type. Then if Ψ and some quasi-inverse to Ψ preserve Frobenius degrees, then they preserve linear morphisms, hence morphisms of Frobenius type [cf. Proposition 1.7, (iii)] and base-isomorphisms [i.e., morphisms of Frobenius type, since C 1 , C 2 are of group-like and isotropic type — cf. Propositions 1.4, (i); 1.7, (ii); 1.8, (iii)]. One way to understand the meaning of the conditions imposed in the various portions of Theorem 3.4 is by considering examples in which some of the conditions hold, but others do not. Example 3.5. Base Categories with FSMI-endomorphisms. Let D be a one-object category whose unique object has endomorphism monoid F; C a one- object category whose unique object has endomorphism monoid F × F. Thus, the projection F × F → F to the ﬁrst factor determines a functor C → D; C may be identiﬁed with the elementary Frobenioid determined by the [manifestly non- dilating] monoid on D that assigns to the unique object of D the monoid Z ≥0 and to every morphism of D the identity automorphism of Z ≥0 . In particular, C is a Frobenioid of Frobenius-normalized and isotropic type, which is not of group-like type [cf. Proposition 1.5, (i), (ii)]. On the other hand, one veriﬁes immediately that every morphism of D is an FSM-morphism, and that the endomorphism 1 ∈ Z ≥0 ⊆ F of the unique object of D is irreducible. Thus, D admits an FSMI-endomorphism, which implies [cf. §0] that D fails to be of FSMFF-type. Moreover, the self- equivalence of C determined by the automorphism of monoids ∼ F × F → F × F 70 SHINICHI MOCHIZUKI given by switching the two factors clearly fails to preserve pre-steps [cf. Theorem 3.4, (ii)]. Example 3.6. Frobenioids of Standard and Group-like Type. Let def Z/pZ G = Z ⊕ p∈Primes [regarded as an abelian group]; D a one-object category whose unique object has endomorphism monoid F G ; C a one-object category whose unique object has endo- morphism monoid F G × F G . Thus, if A ∈ Ob(D), then each automorphism of an object of D arising from an element Aut(D A → D) is contained in the subgroup of inﬁnitely divisible elements of G [cf. the proof of Proposition 1.13, (iii)], hence is trivial — that is to say, D is slim. Moreover, the projection F G × F G → F G to the ﬁrst factor determines a functor C → D; C may be identiﬁed with the elementary Frobenioid determined by the [manifestly non-dilating] monoid on D that assigns to the unique object of D the monoid G and to every morphism of D the identity auto- morphism of G. In particular, C is a Frobenioid of Frobenius-normalized, isotropic, and group-like type [cf. Proposition 1.5, (i), (iii)]. One veriﬁes immediately that every morphism of D is either an isomorphism or a non-monomorphism [cf. the existence of the torsion subgroup p∈Primes Z/pZ ⊆ G], and that the irreducible morphisms of D are precisely the morphisms that project via the natural surjection F G → N ≥1 to primes of N ≥1 . Thus, it follows immediately that D is of FSM-, hence also of FSMFF-type. Moreover, since G pf ∼ = Q = 0, and the ﬁrst factor of F G in the product F G × F G commutes with the G [i.e., “O × (−)”] of the second factor of F G , it follows that the unique object of C is Frobenius-compact. Thus, C is of standard type. On the other hand, the self-equivalence of C determined by the automorphism of monoids ∼ F G × F G → F G × F G given by switching the two factors clearly fails to preserve base-isomorphisms [cf. Theorem 3.4, (iii)]. Example 3.7. Dilating Monoids. Let G, D be as in Example 3.6; Φ the monoid on D that associates to the unique object of D the monoid G × Z ≥0 and to a morphism f ∈ F G of D that projects to an element d f ∈ N ≥1 the endomorphism of G × Z ≥0 that acts trivially on G and by multiplication by d f on Z ≥0 . Thus, [as observed in Example 3.6] D is of FSMFF-type, but Φ clearly fails to be non-dilating. def Write C = F Φ . Thus, C is a Frobenioid of Frobenius-normalized and isotropic type, which is not of group-like type [cf. Proposition 1.5, (i), (ii)]. Moreover, C is a one- object category whose unique object has endomorphism monoid M given by the product set Z ≥0 × (F G × F G ) equipped with the following monoid structure: If a 1 , a 2 ∈ Z ≥0 ; b 1 , b 2 ∈ F G × F G , where b 1 projects to an element (n, m) ∈ N ≥1 × N ≥1 , then (a 1 , b 1 ) · (a 2 , b 2 ) = (a 1 + n · m · a 2 , b 1 · b 2 ) THE GEOMETRY OF FROBENIOIDS I 71 [cf. the description of elementary Frobenioids in Deﬁnition 1.1, (iii)]. Thus, by switching the two factors of F G , and keeping the unique factor of Z ≥0 ﬁxed, we obtain an automorphism of the monoid M , hence a self-equivalence of C, that preserves pre-steps [cf. Theorem 3.4, (ii)], but fails to preserve base-isomorphisms [cf. Theorem 3.4, (iii)]. Example 3.8. ∼ Permutation of Primes. Let α : N ≥1 → N ≥1 be an automor- def def def phism of monoids of order 2; N = (N ≥1 ) gp [so α acts on N ]; U = Q; V = Q; def def W = Q; G = U N , where we let n ∈ N (⊆ Q) act on U by n −1 ; D the one-object category whose unique object has endomorphism monoid G; Φ the [manifestly non- dilating] monoid on D that associates to the unique object of D the monoid V × W and to a morphism g ∈ G that projects to an element n ∈ N the automorphism of V × W given by (α(n), α(n) · n −1 ) [i.e., the automorphism that acts on V by α(n) def and on W by α(n) · n −1 ]; C = F Φ . Thus, C is a Frobenioid of Frobenius-normalized, isotropic, and group-like type [cf. Proposition 1.5, (i), (iii)]; the category D is man- ifestly of FSM-, hence also of FSMFF-type [cf. §0]. Since the unique object of C has “O × (−)” equal to V × W , it follows from our deﬁnition of Φ that this object is Frobenius-compact. Thus, C is of standard type. On the other hand, if A ∈ Ob(D), then Aut(D A → D) ∼ = G; since there exist injections of monoids F → G, it thus follows that D fails to be Frobenius-slim. The monoid M of endomorphisms of the unique object of C may be described as the product set U × V × W × N × N ≥1 equipped with the following monoid structure: if u 1 , u 2 ∈ U ; v 1 , v 2 ∈ V ; w 1 , w 2 ∈ W ; n 1 , n 2 ∈ N ; m 1 , m 2 ∈ N ≥1 , then (u 1 ,v 1 , w 1 , n 1 , m 1 ) · (u 2 , v 2 , w 2 , n 2 , m 2 ) = −1 (u 1 + n −1 1 · u 2 , v 1 + m 1 · α(n 1 ) · v 2 , w 1 + m 1 · α(n 1 ) · n 1 · w 2 , n 1 · n 2 , m 1 · m 2 ) [cf. the description of elementary Frobenioids in Deﬁnition 1.1, (iii)]. In particular, a routine veriﬁcation reveals that the assignment (u, v, w, n, m) → (v, u, w, α(n) −1 · m −1 , α(m)) [where u ∈ U , v ∈ V , w ∈ W , n ∈ N , m ∈ N ≥1 ] determines an automorphism of the monoid M , hence a self-equivalence of C, which clearly preserves base-isomorphisms, but fails to preserve “O × (−)” [i.e., the subspace {0} × V × W ⊆ U × V × W ] or Frobenius degrees [when α is not equal to the identity] — cf. Theorem 3.4, (iii), (iv). Example 3.9. def def def def Non-preservation of Units. Let N = (N ≥1 ) gp ; U = Q; def V = Q; W = Z ≥0 ; G = U N , where we let n ∈ N (⊆ Q) act on U by n −1 ; D the one-object category whose unique object has endomorphism monoid G; Φ the [manifestly non-dilating] monoid on D that associates to the unique object of 72 SHINICHI MOCHIZUKI D the monoid V × W and to a morphism g ∈ G that projects to an element n ∈ N the automorphism of V × W given by (n, 1) [i.e., the automorphism that acts on def V by n and on W by 1]; C = F Φ . Thus, C is a Frobenioid of Frobenius-normalized and isotropic type, which is not of group-like type [cf. Proposition 1.5, (i), (ii)]; D is manifestly of FSM-, hence also of FSMFF-type [cf. §0]. Thus, C is of standard type. On the other hand, [cf. Example 3.8] D fails to be Frobenius-slim. The monoid M of endomorphisms of the unique object of C may be described as the product set U × V × W × N × N ≥1 equipped with the following monoid structure: if u 1 , u 2 ∈ U ; v 1 , v 2 ∈ V ; w 1 , w 2 ∈ W ; n 1 , n 2 ∈ N ; m 1 , m 2 ∈ N ≥1 , then (u 1 ,v 1 , w 1 , n 1 , m 1 ) · (u 2 , v 2 , w 2 , n 2 , m 2 ) = (u 1 + n −1 1 · u 2 , v 1 + m 1 · n 1 · v 2 , w 1 + m 1 · w 2 , n 1 · n 2 , m 1 · m 2 ) [cf. the description of elementary Frobenioids in Deﬁnition 1.1, (iii)]. In particular, a routine veriﬁcation reveals that the assignment (u, v, w, n, m) → (v, u, w, n −1 · m −1 , m) [where u ∈ U , v ∈ V , w ∈ W , n ∈ N , m ∈ N ≥1 ] determines an automorphism of the monoid M , hence a self-equivalence of C, which clearly fails to preserve “O × (−)”, “O (−)” [i.e., the subspaces {0} × V × {0}, {0} × V × W ⊆ U × V × W ] — cf. Theorem 3.4, (iv). Example 3.10. Non-slim Base Categories. Let G be a group, whose center we denote by Z(G); D a one-object category whose unique object has endomorphism monoid G; C a one-object category whose unique object has endomorphism monoid G × F. Thus, the projection G × F → G determines a functor C → D; C may be identiﬁed with the elementary Frobenioid determined by the [manifestly non- dilating] monoid on D that assigns to the unique object of D the monoid Z ≥0 and to every morphism of D the identity automorphism of Z ≥0 . In particular, C is a Frobenioid of Frobenius-normalized and isotropic type, which is not of group-like type [cf. Proposition 1.5, (i), (ii)]; D is manifestly of FSM-, hence also of FSMFF- type [cf. §0]. Thus, C is of standard type. On the other hand, if α : F → Z(G) is any nontrivial homomorphism of monoids that factors as the composite of the natural surjection F → N ≥1 with a homomorphism of monoids N ≥1 → Z(G), then the automorphism of monoids ∼ G × F → G × F (g, f ) → (g · α(f ), f ) ∼ [where g ∈ G, f ∈ F] determines a self-equivalence C → C which clearly fails to preserve base-identity endomorphisms of Frobenius type [cf. Theorem 3.4, (v)]. THE GEOMETRY OF FROBENIOIDS I 73 Finally, before proceeding, we consider the case of Frobenioids of group-like type in a bit more detail. Proposition 3.11. (Frobenioids of Isotropic, Unit-trivial, and Group- like Type) For i = 1, 2, let Φ i be the zero monoid [more precisely: any functor D i → Mon all of whose values are monoids of cardinality one] on a connected, totally epimorphic category D i of FSMFF-type; C i → F Φ i a Frobenioid of isotropic, unit-trivial, and group-like type; ∼ Ψ : C 1 → C 2 an equivalence of categories. Then: (i) The functor C i → F Φ i is an equivalence of categories. (ii) Ψ preserves base-isomorphisms, pull-back morphisms, linear mor- phisms, and morphisms of Frobenius type. (iii) Suppose that both Ψ and some quasi-inverse to Ψ preserve base-identity endomorphisms. Then there exists a 1-unique functor Ψ Base : D 1 → D 2 that ﬁts into a 1-commutative diagram C 1 ⏐ ⏐ D 1 Ψ −→ Ψ Base −→ C 2 ⏐ ⏐ D 2 [where the vertical arrows are the natural projection functors; the horizontal ar- rows are equivalences of categories]. Finally, if D 1 , D 2 are slim, then each of the composite functors of this diagram is rigid. Proof. First, we consider assertion (i). By Proposition 3.3, (iii), (iv), the functor C i → F Φ i is essentially surjective and faithful. Since the Frobenioid C i is of group- like and isotropic type, it follows that every pre-step of C i is an isomorphism [cf. Propositions 1.4, (i); 1.8, (iii)], hence that the Frobenioid C i is of Aut-ample and base-trivial [cf. Deﬁnition 1.3, (i), (b)], as well as unit-trivial, type. Thus, it follows from Proposition 3.3, (v), that the functor C i → F Φ i is an equivalence of categories. This completes the proof of assertion (i). Next, we consider assertion (ii). Observe that since D i is of FSMFF-type, it follows that D i has no FSMI-endomorphisms [cf. §0], hence that a morphism of C i is an FSMI-endomorphism if and only if it is a prime-Frobenius endomorphism [cf. Propositions 1.11, (vi); 1.14, (i); the evident structure of F Φ i ]. Thus, Ψ preserves the prime-Frobenius endomorphisms, hence also prime-Frobenius morphisms [since every prime-Frobenius morphism is abstractly equivalent to a prime-Frobenius en- domorphism]. But this implies that Ψ preserves the morphisms of Frobenius type [cf. Proposition 1.10, (v)], hence also the linear morphisms [cf. Proposition 1.7, (iii)]. Since the [co-angular] pre-steps of C i are isomorphisms [cf. Proposition 1.8, 74 SHINICHI MOCHIZUKI (iii)], it thus follows that Ψ preserves the pull-back morphisms [cf. Proposition 1.7, (iii)], as well as the base-isomorphisms [cf. Proposition 1.7, (ii)]. This completes the proof of assertion (ii). Finally, we consider assertion (iii). Write N for the one-object category whose unique object has endomorphism monoid equal to N ≥1 . Then we have equivalences of categories ∼ ∼ C i → F Φ i → D i × N [cf. assertion (i)]. Moreover, one veriﬁes immediately that the base-identity endo- ∼ morphisms of C i are precisely the endomorphisms of C i → D i × N that arise from elements of N ≥1 ; let us refer to such endomorphisms as “N ≥1 -endomorphisms”. Thus, it follows from our assumption concerning the preservation of base-identity endomorphisms that the N ≥1 -endomorphisms are preserved by Ψ. Note, more- over, that D i may be reconstructed from C i by considering equivalence classes of morphisms of C i , where two morphisms of C i are regarded as equivalent if they admit composites with an N ≥1 -endomorphism which are equal. Thus, we obtain a 1-commutative diagram as in the statement of assertion (ii). Finally, the rigidity assertion in the statement of assertion (ii) follows immediately from Proposition 1.13, (i). THE GEOMETRY OF FROBENIOIDS I 75 Section 4: Category-theoreticity of the Divisor Monoid In the present §4, we show that the monoid on the base category that appears in the deﬁnition of a Frobenioid [cf. Deﬁnition 1.3] may, under suitable conditions, be reconstructed entirely category-theoretically. Together with the results of §3, this allows us to conclude, under suitable conditions, that the functor to an elementary Frobenioid that appears in the deﬁnition of a Frobenioid [cf. Deﬁnition 1.3] may be recovered entirely from the structure of a Frobenioid as an abstract category [cf. Corollary 4.11]. In the following discussion, we maintain the notation of §1, §2, §3. Also, we assume that we have been given a divisorial monoid Φ on a connected, totally epimorphic category D and a Frobenioid C → F Φ . Proposition 4.1. (Primary Steps) Suppose further that C is of perfect and isotropic type, and that Φ is perf-factorial. Let A ∈ Ob(C) be Div-Frobenius- trivial; φ : B → A, ψ : A → C, δ : D → E, : E → F, ι : I → F steps of C. For n ∈ N ≥1 , let α n ∈ End C (A) be a Div-identity endomorphism of Frobenius type such that deg Fr (α n ) = n. Then: (i) φ is primary if and only if, for every factorization φ = φ A ◦ φ B , where φ B : B → B , φ A : B → A are steps, there exists a commutative diagram B φ B −→ φ A B −→ ⏐ ⏐ β B ζ −→ A ⏐ ⏐ α n A where n ∈ N ≥1 ; β is a morphism of Frobenius type; and ζ = φ◦ζ ; and ζ : B → B is a pre-step. (ii) Suppose that φ is primary. Then the composite ψ ◦ φ : B → C, hence also ψ, is primary if and only if, for every factorization ψ ◦ φ = ψ ◦ φ , where φ : B → A , ψ : A → C are steps, there exist factorizations φ = ζ ◦φ , φ = ζ ◦φ , where φ : B → A is a step, and ζ : A → A, ζ : A → A are pre-steps. (iii) ∗ (Div( )), ι ∗ (Div(ι)) ∈ Φ(F ) [where we write ∗ , ι ∗ for the respective bijections induced by the functor Φ] have disjoint supports [cf. Deﬁnition 2.4, (i), (d)] if and only if every pre-step ζ : Z → F such that there exist pre-steps , ι satisfying = ζ ◦ , ι = ζ ◦ ι is, in fact, an isomorphism. In this case, we shall say that , ι are co-primary. If , ι are co-primary, then there exists a cartesian diagram in the category of pre-steps U −→ ⏐ ⏐ ι I ι −→ E ⏐ ⏐ F 76 SHINICHI MOCHIZUKI such that ∗ ( ∗ (Div( ))) = ι ∗ (Div(ι)), ι ∗ (ι ∗ (Div(ι ))) = mary, then so are , ι . ∗ (Div( )); if , ι are pri- (iv) δ is primary if and only if there exists a p ∈ Prime(Φ(F )) such that the following condition holds: For every primary : E → F such that ∗ (Div( )) ∈ p [where we write ∗ for the bijection induced by the functor Φ], there exists a factorization = ◦ζ, where ζ is a pre-step, if and only if there exists a factorization ◦ δ = ◦ θ, where θ is a pre-step. (v) is primary if and only if there exists a p ∈ Prime(Φ(D)) such that the following condition holds: For every primary δ : D → E such that Div(δ ) ∈ p, there exists a factorization δ = ζ ◦ δ , where ζ is a pre-step, if and only if there exists a factorization ◦ δ = θ ◦ δ , where θ is a pre-step. Proof. First, we consider assertion (i). By applying the second equivalence of categories of Deﬁnition 1.3, (iii), (d), to the various pre-steps over A, it follows def that, if we write x φ = φ ∗ (Div(φ)) ∈ Φ(A) [where we write φ ∗ for the bijection induced by the functor Φ], then the condition of assertion (i) may be translated into the language of monoids as follows: For every equation x φ = x A + x B in Φ(A), where x A , x B = 0, we have x φ x A . Now the equivalence of this condition with the condition that x φ is primary follows immediately from the deﬁnition of the term “primary” [cf. §0], together with the fact that Φ(A) is perfect [cf. Proposition 1.10, (iii)]. This completes the proof of assertion (i). Next, we consider assertion (ii). Again, we apply Deﬁnition 1.3, (iii), (d), to the various pre-steps over C, to obtain the following translation of the condition of as- def def sertion (ii) into the language of monoids [where we set x φ = ψ ∗ (φ ∗ (Div(φ))), x ψ = ψ ∗ (Div(ψ)) ∈ Φ(C)]: For every equation x φ + x ψ = x φ + x ψ in Φ(A), where x φ , x ψ = 0, there exists a 0 = x φ ∈ Φ(A) such that x φ ≤ x φ , x φ ≤ x φ . Now the necessity of this condition follows immediately from the structure of the Φ(A) p , where p ∈ Prime(Φ(A)) [cf. Deﬁnition 2.4, (i), (b)], whereas the suﬃciency of this condition follows by taking x φ ≤ x ψ [cf. Deﬁnition 2.4, (i), (c), (d); the fact that Φ(A) is perfect]. This completes the proof of assertion (ii). Next, we consider assertion (iii). By applying the second equivalence of cat- egories of Deﬁnition 1.3, (iii), (d), to the various pre-steps over F , we obtain the following translation of the condition of assertion (iii) into the language of monoids def def [where we set x = ∗ (Div( )), x ι = ι ∗ (Div(ι)) ∈ Φ(F )]: Every x ζ ∈ Φ(F ) such that x ζ ≤ x , x ζ ≤ x ι is, in fact, equal to 0. THE GEOMETRY OF FROBENIOIDS I 77 The necessity and suﬃciency of this condition then follow immediately by consid- ering the “primary factorizations” of x , x ι [cf. Deﬁnition 2.4, (i), (c), (d); the fact that Φ(A) is perfect]. The cartesian diagram [with the desired properties] then follows from the fact that “for x U ∈ Φ(F ), x + x ι ≤ x U if and only if x ≤ x U , x ι ≤ x U ” [cf. Deﬁnition 2.4, (i), (c), (d); the fact that Φ(A) is perfect]. This completes the proof of assertion (iii). Next, we consider assertion (iv). This time, we apply the second equivalence of categories of Deﬁnition 1.3, (iii), (d), to the various pre-steps over F , to obtain the following translation of the condition of assertion (iv) concerning p ∈ Prime(Φ(F )) def into the language of monoids [where we set x δ = Φ(F )]: ∗ (δ ∗ ((Div(δ))), x def = ∗ (Div( )) ∈ / p, x ≤ x if and only if x ≤ x δ + x . For every primary element x ∈ The necessity and suﬃciency of this condition then follow immediately by com- paring the “primary factorizations” of x , x δ + x [cf. Deﬁnition 2.4, (i), (c), (d); the fact that Φ(A) is perfect]. Also, we observe that assertion (v) follows by an entirely similar argument obtained by “reversing the direction of the arrows”. This completes the proof of assertions (iv), (v). Theorem 4.2. (Category-theoreticity of Primary Steps) For i = 1, 2, let Φ i be a perf-factorial divisorial monoid on a connected, totally epimorphic category D i ; C i → F Φ i a Frobenioid of standard and isotropic type, which is not of group-like type; ∼ Ψ : C 1 → C 2 an equivalence of categories. Then: (i) Ψ preserves primary steps, Div-identity endomorphisms, Div-Frobenius- trivial objects, and universally Div-Frobenius-trivial objects. (ii) There exists a unique isomorphism Ψ Prime between the functors Ob(C i bs-iso ) A i → Prime(Φ i (A i )) [where i = 1, 2] on C i bs-iso which satisﬁes the following property: Suppose that A 2 = Ψ(A 1 ); p 1 ∈ Prime(Φ 1 (A 1 )), p 2 ∈ Prime(Φ 2 (A 2 )) correspond under Ψ Prime . For i = 1, 2, write ∼ { A i (C i coa-pre )} p i → Order(Φ i (A i ) p i ); ∼ {(C i coa-pre ) A i } p i → Order(Φ i (A i ) p i ) opp for the respective full subcategories and restricted equivalences of categories deter- mined by the full subcategory Order(Φ i (A i ) p i ) ⊆ Order(Φ i (A i )) 78 SHINICHI MOCHIZUKI arising from the submonoid Φ i (A i ) p i ⊆ Φ i (A i ). Then the map induced by Ψ on pre-steps [cf. (i); Theorem 3.4, (ii)] induces equivalences of categories ∼ { A 1 (C 1 coa-pre )} p 1 → { A 2 (C 2 coa-pre )} p 2 ; ∼ {(C 1 coa-pre ) A 1 } p 1 → {(C 2 coa-pre ) A 2 } p 2 hence equivalences of categories as follows: ∼ Order(Φ 1 (A 1 ) p 1 ) → Order(Φ 2 (A 2 ) p 2 ); ∼ Order(Φ 1 (A 1 ) p 1 ) opp → Order(Φ 2 (A 2 ) p 2 ) opp (iii) If, moreover, in the situation of (ii), the A i are Div-Frobenius-trivial, then the last two equivalences of categories of (ii) arise from isomorphisms of monoids ∼ ∼ Φ 1 (A 1 ) p 1 → Φ 2 (A 2 ) p 2 ; Φ 1 (A 1 ) p 1 → Φ 2 (A 2 ) p 2 which we shall refer to, respectively, as the right-hand and left-hand isomor- phisms induced by Ψ [cf. Example 4.3 below]. Proof. First, we observe that by Proposition 1.10, (vi), every group-like object is Frobenius-trivial, hence, in particular, Div-Frobenius-trivial; moreover, [by the deﬁnition of a “group-like object”] every endomorphism of a group-like object is a Div-identity endomorphism, and every pre-step to or from a group-like object is an isomorphism [cf. Propositions 1.4, (i), (iii); 1.8, (iii)]. Thus, since Ψ pre- serves non-group-like objects [cf. Theorem 3.4, (ii)] and pull-back morphisms [cf. Theorem 3.4, (iii)], we may assume for the remainder of the proof of Theorem 4.2, without loss of generality, that the objects under consideration are non-group-like. Now by Proposition 1.14, (v) [cf. also Theorem 3.4, (ii)], it follows immediately that Ψ preserves non-group-like Div-Frobenius-trivial objects, as well as Div-identity prime-Frobenius endomorphisms of such objects. Since Ψ preserves morphisms of Frobenius type and Frobenius degrees [cf. Theorem 3.4, (iii)], we thus conclude that to complete the proof of assertion (i), it suﬃces to prove that Ψ preserves primary steps and Div-identity endomorphisms. Moreover, to prove the remainder of asser- tion (i) [i.e., that Ψ preserves primary steps and Div-identity endomorphisms] and assertions (ii), (iii), clearly it suﬃces to do so after passing to the perfections of the C i [cf. Theorem 3.4, (iii)]; thus, for the remainder of the proof of Theorem 4.2, we may assume, without loss of generality, that the C i are of perfect type [cf. also Proposition 5.5, (iii), below]. def Now let A 1 ∈ Ob(C 1 ) be a non-group-like Div-Frobenius-trivial object; A 2 = Ψ(A 1 ). Then it follows formally from Proposition 4.1, (i), (ii) [cf. also Theorem 3.4, (ii), (iii)] that Ψ maps primary steps to or from A 1 to primary steps to or from A 2 in such a way that primary steps B 1 → A 1 , A 1 → C 1 with primary composite B 1 → C 1 are mapped to primary steps B 2 → A 2 , A 2 → C 2 with primary composite B 2 → C 2 . Next, let A 1 → F 1 be a primary step. Then it follows immediately from Proposition 4.1, (iii), together with what we have already shown concerning primary steps to or from A 1 , that Ψ def maps primary steps to or from F 1 to primary steps to or from F 2 = Ψ(F 1 ) in such THE GEOMETRY OF FROBENIOIDS I 79 a way that primary steps F 1 → F 1 , F 1 → F 1 with primary composite F 1 → F 1 are mapped to primary steps F 2 → F 2 , F 2 → F 2 with primary composite F 2 → F 2 . [Indeed, to see this, it suﬃces to consider the following two situations [depending on whether the primary steps A 1 → F 1 , F 1 → F 1 are co-primary or not]: (a) primary steps B i → A i , A i → C i with primary composite such that the primary steps to or from F i under consideration are subordinate to the primary composite B i → C i ; (b) commutative diagrams A i −→ F i ⏐ ⏐ ⏐ ⏐ A i ⏐ ⏐ −→ F i ⏐ ⏐ A i −→ F i [where i = 1, 2] in which both the upper and lower squares are cartesian diagrams as in Proposition 4.1, (iii), and all the arrows originating from A i , as well as the vertical composite A i → A i → A i , are primary steps.] Next, observe that for a suitable choice of non-group-like Div-Frobenius-trivial A 1 [e.g., a Frobenius-trivial A 1 — cf. Deﬁnition 1.3, (i), (a), (b)], it follows that for any object C 1 ∈ Ob(C 1 ) that is base-isomorphic to A 1 , there exist pre-steps B 1 → C 1 , B 1 → A 1 . Moreover, observe that [by applying the equivalences of categories of Deﬁnition 1.3, (iii), (d)] any primary step to or from B 1 , as well as any primary composite of a primary step to B 1 with a primary step from B 1 , may always be written in the form D 1 → E 1 where the composite D 1 → E 1 → F 1 of the above arrow D 1 → E 1 with some arrow E 1 → F 1 factors as a composite D 1 → B 1 → A 1 → F 1 in which D 1 → B 1 is a pre- step, B 1 → A 1 is the pre-step introduced above, and A 1 → F 1 is a primary step [so in the case of a primary step from B 1 , D 1 = B 1 ; in the case of a primary step to B 1 , E 1 = B 1 ]. Thus, by applying Proposition 4.1, (iv) [to the arrows D 1 → E 1 → F 1 ], together with what we have already shown concerning primary steps to or from F 1 , we conclude that Ψ maps primary steps to or from B 1 to primary steps to or def from B 2 = Ψ(B 1 ) in such a way that primary steps B 1 → B 1 , B 1 → B 1 with primary composite B 1 → B 1 are mapped to primary steps B 2 → B 2 , B 2 → B 2 with primary composite B 2 → B 2 . In a similar vein, we observe that [by applying the equivalences of categories of Deﬁnition 1.3, (iii), (d)] a primary step to or from C 1 , as well as any primary composite of a primary step to C 1 with a primary step from C 1 , may always be written in the form E 1 → F 1 where the composite D 1 → E 1 → F 1 of some arrow D 1 → E 1 with the above arrow E 1 → F 1 factors as a composite D 1 → B 1 → C 1 → F 1 in which D 1 → B 1 is a primary pre-step, B 1 → C 1 is the pre-step introduced above, and C 1 → F 1 is a pre-step [so in the case of a primary step from C 1 , E 1 = C 1 ; in the case of 80 SHINICHI MOCHIZUKI a primary step to C 1 , F 1 = C 1 ]. Thus, by applying Proposition 4.1, (v) [to the arrows D 1 → E 1 → F 1 ], together with what we have already shown concerning primary steps to or from D 1 [i.e., where we regard “D 1 ” as a “sort of B 1 ”, which is possible in light of the existence of the composite pre-step D 1 → B 1 → A 1 ], we conclude that Ψ maps primary steps to or from C 1 to primary steps to or from def C 2 = Ψ(C 1 ) in such a way that primary steps C 1 → C 1 , C 1 → C 1 with primary composite C 1 → C 1 are mapped to primary steps C 2 → C 2 , C 2 → C 2 with primary composite C 2 → C 2 . Since C 1 was, in eﬀect, allowed to be an arbitrary non-group-like object of C 1 , we thus conclude that Ψ preserves primary steps. Moreover, by thinking, for A i ∈ Ob(C i ) [where i = 1, 2] of an element of Prime(Φ i (A i )) as an equivalence class of primary steps to or from A i [where the correspondence between elements of Prime(Φ i (A i )) and equivalence classes of primary steps is deﬁned by “Div(−)” — cf. the equivalences of categories of Deﬁnition 1.3, (iii), (d)], we thus obtain that Ψ induces a bijection ∼ Ψ Prime (A 1 ) : Prime(Φ 1 (A 1 )) → Prime(Φ 2 (A 2 )) def [where A 2 = Ψ(A 1 )] as well as corresponding equivalences of categories ∼ { A 1 (C 1 coa-pre )} p 1 → { A 2 (C 2 coa-pre )} p 2 ; ∼ {(C 1 coa-pre ) A 1 } p 1 → {(C 2 coa-pre ) A 2 } p 2 [where p 1 ∈ Prime(Φ 1 (A 1 )), p 2 ∈ Prime(Φ 2 (A 2 )) correspond via Ψ Prime (A 1 )]. To check the functoriality of Ψ Prime (−) with respect to arbitrary base-isomor- phisms, it suﬃces to check it with respect to morphisms of Frobenius type and pre- steps [cf. Proposition 1.7, (ii)]. In the case of a morphism of Frobenius type B i → A i [where i = 1, 2], the desired functoriality follows by considering commutative diagrams B i −→ B i ⏐ ⏐ ⏐ ⏐ A i −→ A i [cf. Proposition 1.10, (i)] — where the vertical morphisms are morphisms of Frobe- nius type, and the horizontal morphisms are primary steps. In the case of a pre-step B i → A i [where i = 1, 2], the desired functoriality follows by considering a commu- tative diagram B i −→ B i ⏐ ⏐ ⏐ ⏐ C i ⏐ ⏐ −→ C i ⏐ ⏐ A i −→ A i — where all of the morphisms are pre-steps; all of the horizontal morphisms, as well as the vertical morphisms and composite morphisms of the upper square, are either THE GEOMETRY OF FROBENIOIDS I 81 isomorphisms or primary steps; either the vertical morphisms of the lower square are isomorphisms, or the lower square is a cartesian diagram as in Proposition 4.1, (iii). This completes the proof of the functoriality of Ψ Prime (−), hence of assertion (ii). Next, we observe that Ψ preserves Div-identity endomorphisms. Indeed, since the Φ i are non-dilating, it follows that if A ∈ Ob(C i ) [where i = 1, 2], then α ∈ End C i (A) is a Div-identity endomorphism if and only if α admits a factorization α = β ◦ γ, where β : B → A is a pull-back morphism, and γ : A → B is a base- isomorphism, such that for every primary step A → A, there exists a commutative diagram A −→ A ⏐ ⏐ ⏐ γ ⏐ γ B −→ ⏐ ⏐ B ⏐ ⏐ β A A β −→ in which the horizontal morphisms are primary steps; the upper horizontal mor- phism is the given primary step; the equivalence classes of the primary steps A → ∼ A, B → B correspond via the bijection Prime(Φ i (γ)) : Prime(Φ i (B)) → Prime(Φ i (A)) [cf. the functoriality of Ψ Prime (−)]; β is a pull-back morphism [cf. Proposition 1.11, (v)]; the primary steps A → A, A → A determine the same element of Prime(Φ i (A)). This completes the proof of assertion (i). Finally, we consider assertion (iii). Thus, we assume that the A i are Div- Frobenius-trivial. By considering commutative diagrams of the form B i −→ A i A i −→ C i ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ B i −→ A i A i −→ C i — where the vertical morphisms are morphisms of Frobenius type [cf. Proposition 1.10, (i)], the morphisms A i → A i are Div-identity endomorphisms, and the hori- zontal morphisms are primary steps — it follows that the equivalences of categories in question arise from bijections of sets ∼ Φ 1 (A 1 ) p 1 → Φ 2 (A 2 ) p 2 ; ∼ Φ 1 (A 1 ) p 1 → Φ 2 (A 2 ) p 2 that are compatible both with “≤” and with multiplication by elements of N ≥1 . In light of the well-known structure of the monoids Q ≥0 , R ≥0 [cf. Deﬁnition 2.4, (i), (b)], this is enough to conclude that these bijections of sets are, in fact, isomor- phisms of monoids, as desired. This completes the proof of assertion (iii). Example 4.3. Independence of Right-hand and Left-hand Isomor- phisms. As the following example shows, the right-hand and left-hand isomor- phisms of Theorem 4.2, (iii), do not necessarily coincide [cf. Remark 4.9.1 below]: 82 SHINICHI MOCHIZUKI Let D be a one-morphism category; Φ the monoid on D whose value on the unique object of D is the monoid Q ≥0 . Now we deﬁne a category C as follows: The objects of C are the elements of Q. The morphisms a → b of C from an object a ∈ Q to an object b ∈ Q are the elements d ∈ N ≥1 such that d · a ≤ b; composition of morphisms is deﬁned by multiplication of elements of N ≥1 . We shall refer to the element d ∈ N ≥1 determined by a morphism of C as the Frobenius degree of the morphism. Thus, we obtain a natural functor C → F Φ by assigning to a morphism φ : a → b [where a, b ∈ Q] the zero divisor b − deg Fr (φ) · a ∈ Q ≥0 and Frobenius degree deg Fr (φ) ∈ N ≥1 . Since C is clearly connected and totally epimorphic, this functor determines a pre-Frobenioid structure on C. More- over, the object 0 ∈ Q is Frobenius-trivial; φ : a → b is a morphism of Frobenius type if and only if b = deg Fr (φ) · a; φ : a → b is a pre-step if and only if deg Fr (φ) = 1; all morphisms of C are base-isomorphisms; all pull-back morphisms of C are iso- morphisms; all “O (−)” of C are trivial; no object of C is group-like. Thus, one veriﬁes immediately that C is a Frobenioid of isotropic type. Since D is clearly of FSMFF-type, and Φ is non-dilating, it follows that C is also of standard type, over a slim base category D. Now one veriﬁes immediately that if λ ∈ Q >0 , then the assignment, for a ∈ Q ≥0 , a → a; −a → −λ · a determines a self-equivalence of categories ∼ Ψ λ : C → C that preserves Frobenius degrees [cf. Theorem 3.4, (iii)]. On the other hand, it follows immediately from the construction of Ψ λ that the right-hand isomorphism of Theorem 4.2, (iii), is the identity on Q ≥0 , while the left-hand isomorphism of Theorem 4.2, (iii), is given by multiplication by λ on Q ≥0 . In order to proceed further toward the goal of “reconstructing Φ category- theoretically from C”, it is necessary to ﬁnd natural conditions on the Frobenioid C that will allow us to rule out “pathologies” of the sort discussed in Example 4.3. One approach to doing this is the introduction of the birationalization of a Frobenioid, as follows. Proposition 4.4. (Birationalization of a Frobenioid I) For A, B ∈ Ob(C), write: def (A, B) = lim Hom C (A , B) Hom birat C − → coa-pre (A →A)∈C A where the inductive limit is parametrized by [say, some small skeletal subcat- coa-pre egory of ] C A , and the transition morphism induced by a co-angular pre-step coa-pre ] is the natural morphism Hom C (A , B) → A → A [regarded as a morphism in C A Hom C (A , B). Then: THE GEOMETRY OF FROBENIOIDS I 83 (i) Composition of morphisms in C determines a natural composition map (A, B) × Hom birat (B, C) → Hom birat (A, C) Hom birat C C C [where A, B, C ∈ Ob(C)], hence a category C birat , whose objects are the objects of ”. Moreover, there exists a natural C and whose morphisms are given by “Hom birat C 1-commutative diagram of functors C ⏐ ⏐ −→ F Φ ⏐ ⏐ C birat −→ F Φ gp −→ F 0 D where the functors between elementary Frobenioids are those induced by the natural morphisms of monoids Φ → Φ gp → 0 D ; 0 D is the monoid on D all of whose values on objects of D are equal to the monoid with one element [so F 0 D is the product category of D with the one-object category determined by the monoid N ≥1 ]. (ii) The functor C birat → F 0 D of (i) determines a structure of Frobenioid of group-like type on C birat . Moreover, the functor C → C birat is faithful. In particular, for every A ∈ Ob(C) with image A birat in C birat , the functor C → C birat determines an injection of groups O (A) gp → O × (A birat ). We shall refer to the functor “O × (−)” on D associated to the Frobenioid C birat [cf. Proposition 2.2, (ii), (iii)] as the rational function monoid of the Frobenioid C. (iii) There exists a unique subfunctor of groups Φ birat ⊆ Φ gp such that the functor C birat → F Φ gp of (i) factors through the subcategory F Φ birat ⊆ F Φ gp determined by Φ birat , and, moreover, the resulting functor C birat → F Φ birat induces, for each A birat ∈ Ob(C birat ), a surjection O × (A birat ) Φ birat (A birat ), whose kernel is the image, via the injection O (A) gp → O × (A birat ) of (ii), of O × (A) ⊆ O (A) gp . (iv) A morphism of C maps to a(n) co-angular morphism (respectively, iso- morphism; morphism of Frobenius type; pull-back morphism; morphism of a given Frobenius degree; isometry; pre-step; base-isomorphism) of C birat if and only if it is a(n) co-angular morphism (respectively, co-angular pre- step; co-angular base-isomorphism; co-angular linear morphism; morphism of a given Frobenius degree; arbitrary morphism; pre-step; base-isomorphism) of C. A morphism of C birat is a base-identity endomorphism if and only if arises from a pair (α : A → A; φ : A → A), where α is a co-angular pre-step in the in- dexing category of the inductive limit deﬁning Hom birat (A, A), and α and φ are C base-equivalent. An object of C maps to an isotropic object of C birat if and only if it is an isotropic object of C. 84 SHINICHI MOCHIZUKI Proof. First, we consider assertion (i). Given morphisms φ : A → B, ψ : B → C [in C] and co-angular pre-steps α : A → A, β : B → B [in C], it follows from Proposition 1.11, (vii), that there exists a commutative diagram φ ψ A −→ B −→ C ⏐ ⏐ ⏐ β ⏐ α α A ←− A φ −→ B where α [hence also α ◦ α ] is a co-angular pre-step. Then we take the composite of (A, B) with the image of ψ in Hom birat (B, C) to be the the image of φ in Hom birat C C birat image of ψ ◦ φ in Hom C (A, C). To show that this assignment is independent of the choice of α , φ , it suﬃces to consider commutative diagrams α ∗ A ∗ −→ ⏐ ⏐ α A α −→ φ A −→ B ⏐ ⏐ ⏐ ⏐ β α A φ −→ B [where α , α , α ∗ are co-angular pre-steps] and to observe that since β is a monomorphism [cf. Deﬁnition 1.3, (v), (a)], the fact that β ◦ φ ◦ α ∗ = φ ◦ α ◦ α ∗ = φ ◦ α ◦ α = β ◦ φ ◦ α implies that φ ◦α ∗ = φ ◦α , i.e., that ψ ◦φ , ψ ◦φ determine the same element (A, C). Also, one veriﬁes immediately that composite of morphisms of of Hom birat C Hom birat (−, −) is associative. This completes the deﬁnition of the category C birat . C Then by assigning to the pair (α : A → A, φ : A → B) the element Φ(α) −1 {Div(φ ) − deg Fr (φ ) · Div(α)} ∈ Φ(A) gp [cf. Remark 1.1.1] one veriﬁes immediately that the functor C → F Φ induces a functor C birat → F Φ gp , as well as a 1-commutative diagram as in the statement of assertion (i). This completes the proof of assertion (i). Next, we observe that it follows formally from the deﬁnition of C birat that C birat is connected; moreover, [cf. the discussion of the composition of arrows of C birat in the proof of assertion (i)] the total epimorphicity of C birat follows immediately from that of C. Thus, the functor C birat → F 0 D determines a structure of pre-Frobenioid on C birat . Now the portion of assertion (iv) concerning morphisms of a given Frobe- nius degree, isometries [cf. the monoid structure of the monoid 0 D !], pre-steps, base-isomorphisms, and base-identity endomorphisms of C birat follows immediately from the deﬁnitions. The portion of assertion (iv) concerning co-angular pre-steps (−, −)”; Proposition 1.7, of C follows immediately from the deﬁnition of “Hom birat C (v) [for co-angular pre-steps]. To verify the portion of assertion (iv) concerning co-angular morphisms, we reason as follows: Given a morphism A birat → B birat in C birat , any factorization THE GEOMETRY OF FROBENIOIDS I 85 A birat → C birat → D birat → B birat in C birat , where either A birat → C birat or D birat → B birat is a base-isomorphism, C birat → D birat is a(n) [isometric] pre- step, and D birat → B birat is linear, arises [cf. the proof of assertion (i)] from a factorization A → C → D → B in C, where either A → C or D → B is a base-isomorphism, C → D is a pre-step, and D → B is linear. Thus, if A → B is co-angular, then [by applying the factorization of Deﬁnition 1.3, (v), (b), to C → D , we conclude that] C → D is a co-angular pre-step, so C birat → D birat is an isomorphism; in particular, it follows that A birat → B birat is co-angular. On the other hand, if A birat → B birat is co-angular, then C birat → D birat is an isomorphism, which [by the portion of assertion (iv) concerning isomorphisms of C birat ] implies that C → D is a co-angular pre-step, hence an isomorphism whenever it is an isometry [cf. Proposition 1.4, (iii)]; thus, A → B, hence also any morphism A → B appearing in a factorization A → A → B [where A → A is a co-angular pre-step], is co-angular. The portion of assertion (iv) concerning morphisms of Frobenius type now follows formally from the portion of assertion (iv) concerning co-angular morphisms, isometries, and base-isomorphisms. Next, let us observe that it is immediate from the deﬁnition of a pull-back morphism [cf. Deﬁnition 1.2, (ii)] that any pull-back morphism of C maps to a pull-back morphism of C birat . Since, moreover, a morphism of C is a co-angular linear morphism if and only if it is a composite of a co-angular pre-step and a pull-back morphism [cf. Propositions 1.4, (iv); 1.7, (iii)], it thus follows [cf. the portion of assertion (iv) concerning co-angular pre-steps of C] that every co-angular linear morphism of C maps to a pull-back morphism of C birat . On the other hand, if φ : A → B is a morphism of C that maps to a pull-back morphism φ birat : A birat → B birat of C birat , then it follows that φ is linear, hence that it factors as a composite γ ◦ α, where α : A → C is a pre-step, and γ : C → B is a pull-back morphism [cf. Proposition 1.7, (iii)]. Thus, we obtain an equation φ birat = γ birat ◦ α birat in C birat , where φ birat , γ birat are pull-back morphisms, and α birat is a base-isomorphism; but [by the isomorphism of functors appearing in the deﬁnition of a “pull-back morphism” in Deﬁnition 1.2, (ii)] this implies formally that α birat is an isomorphism, hence [by the portion of assertion (iv) concerning co-angular pre-steps of C] that α is a co-angular pre-step, as desired. Finally, the portion of assertion (iv) concerning isotropic objects follows immediately from the portion of assertion (iv) concerning pre-steps and co-angular pre-steps; Proposition 1.4, (i), (iii); Proposition 1.9, (iv). This completes the proof of assertion (iv). In light of the “dictionary” provided by assertion (iv) [cf. also Proposition 1.4, (iv); the equivalence of categories of Proposition 1.9, (ii)], it is now a routine exercise to check that C birat is, in fact, a Frobenioid of group-like type. Moreover, it is immediate from the deﬁnitions [and the total epimorphicity of C] that the functor C → C birat is faithful and determines an injection O (A) gp → O × (A birat ), for A ∈ Ob(C). This completes the proof of assertion (ii). Now assertion (iii) follows immediately from the existence of the functor C birat → F Φ gp of assertion (i) [cf. also Proposition 1.5, (ii)]; here, we note that the computation of the kernel of the surjection of assertion (iii) follows from Deﬁnition 1.3, (vi). 86 SHINICHI MOCHIZUKI Deﬁnition 4.5. (i) We shall say that an object of C is birationally Frobenius-normalized if its image in C birat is Frobenius-normalized. [Thus, any birationally Frobenius- normalized object of C is Frobenius-normalized — cf. Proposition 4.4, (ii), (iv).] If every object of C is birationally Frobenius-normalized, then we shall say that C is of birationally Frobenius-normalized type. If C is of pre-model and birationally Frobenius-normalized type, then we shall say that C is of model type. (ii) Suppose that Φ is perf-factorial; A ∈ Ob(C). Then we shall say that A is strictly rational if, for every prime p ∈ Prime(Φ(A)), there exists an element / Supp(b) [cf. a − b ∈ Φ birat (A), where a, b ∈ Φ(A) such that p ∈ Supp(a), p ∈ Deﬁnition 2.4, (i), (d)]. We shall say that A is rational if there exists a pull-back morphism B → A in C, where B is strictly rational. If [Φ is perf-factorial, and] every object of C is rational (respectively, strictly rational), then we shall say that C is of rational (respectively, strictly rational) type. (iii) We shall say that C is of rationally standard type if the following conditions are satisﬁed: (a) C is of birationally Frobenius-normalized, rational, and standard type; (b) (C un-tr ) birat admits a Frobenius-compact object. (iv) We shall say that D is Div-slim [relative to Φ] if, for every A ∈ Ob(D), the homomorphism Aut(D A → D) → Aut(D A → Mon) [induced by composition with the functor Φ : D → Mon] is injective. [Thus, if D is slim, then it is Div-slim.] Remark 4.5.1. We observe in passing that it is immediate from the deﬁnitions that if C is of rationally standard type (respectively, of standard type), then so is C istr . Example 4.6. Frobenius-normalized vs. Birationally Frobenius-normal- ized. As the following example shows, it is not necessarily the case that a Frobe- nioid of Frobenius-normalized type is of birationally Frobenius-normalized type: Let G be an abelian group, written additively. For each p ∈ Primes, let ξ p ∈ G. def Then if we write M = G × Z × Z, then the assignment M (g, a, b) → (p · g + a · ξ p , p · a, p · b) ∈ M determines an endomorphism α p ∈ End(M ) of the module M such that α p com- mutes with all α p , for p ∈ Primes. Thus, we obtain a homomorphism N ≥1 → End(M ), i.e., an action of N ≥1 on M ; write α n for the image in End(M ) of n ∈ N ≥1 . Write N for the monoid whose underlying set is equal to the direct product M × N ≥1 THE GEOMETRY OF FROBENIOIDS I 87 and whose monoid structure is given as follows: If λ, μ ∈ M ; l, m ∈ N ≥1 , then (λ, l) · (μ, m) = (λ + α l (μ), l · m). Now let D be a one-morphism category; Φ the monoid on D whose unique value is given by Z ≥0 × Z ≥0 . Let C be the category whose objects A n are indexed by elements n ∈ Z, and whose morphisms A n 1 → A n 2 [where n 1 , n 2 ∈ Z] consist of elements (g, a, b, d) ∈ N such that a ≥ 0, b ≥ 0, n 2 −d·n 1 = a+b; composition of morphisms is determined by the product structure of N . The assignment (g, a, b, d) → (a, b, d) then determines a functor C → F Φ [which lies over D]. Moreover, one checks immediately that, relative to this last functor, C is a Frobenioid of isotropic and standard type which is not of group-like type. Also, we observe that the object A 0 ∈ Ob(C) is Frobenius-trivial, and that for every A ∈ Ob(C), O × (A) = O (A) = G. On the other hand, one computes easily that for A birat ∈ Ob(C birat ), O × (A birat ) = M 0 , where we write M 0 ⊆ M for the subgroup of (g, a, b) ∈ M such that a + b = 0. Moreover, the morphisms ) (0, 0, 0, d) ∈ N determine a homomorphism N ≥1 → End C (A 0 ) → End C birat (A birat 0 [where we write use the superscript “birat” to denote the image of objects of C in C birat ], hence an action of N ≥1 on O × (A birat ) = M 0 , which is easily veriﬁed to coincide with the restriction to M 0 of the original action of N ≥1 on M . Now observe that C is of [strictly] rational type [cf. Deﬁnition 4.5, (ii)], and, moreover, every object of (C un-tr ) birat is Frobenius-compact. On the other hand, if the ξ p = 0 [so α p does not act on M 0 by multiplication by p], then C fails to be of birationally Frobenius-normalized type. [In a similar vein, we note that although C birat is “very similar” to an elementary Frobenioid, the presence of the “ξ p ’s” means that it is not, in general, an elementary Frobenioid.] Example 4.7. Frobenius-slim vs. Div-slim. (i) Suppose that the functor Φ : D → Mon maps every automorphism of D to an identity automorphism of Mon. Then it follows formally that D is Div-slim if and only if D is slim. In particular, if, for instance, D is a one-object category, A ∈ Ob(D), and End D (A) is a nontrivial residually ﬁnite group G, then Aut(D A → D) = Ker(Aut(D A → D) → Aut(D A → Mon)) = G — so [cf. Remark 3.1.2] D is Frobenius-slim, but not Div-slim. def def def (ii) Let V = Q; N = (N ≥1 ) gp ; G = V N , where N (⊆ Q) acts on V multi- plicatively. Let D be a one-object category, A ∈ Ob(D); suppose that End D (A) = G [so Aut(D A → D) = G]. Then clearly there exists an injection F → G, so D fails to be Frobenius-slim. On the other hand, if Φ : D → Mon is the functor determined by the monoid Z ≥0 g∈G [i.e., the copies of Z ≥0 are indexed by the elements of G] equipped with the G-action obtained by letting G act by left multiplication on the indices of the copies of Z ≥0 , then the natural map Aut(D A → D) = G → Aut(D A → Mon) 88 SHINICHI MOCHIZUKI is clearly injective, so D is Div-slim [relative to Φ]. Proposition 4.8. (Birationalization of a Frobenioid II) (i) If C is of isotropic type, then so is C birat . (ii) If C is of perfect and isotropic type, then so is C birat . (iii) If C is of rationally standard type, then (C istr ) birat is of standard type. (iv) If C is of isotropic and pre-model type, then so is C birat . Proof. Assertion (i) follows formally from Proposition 4.4, (iv). To prove assertion (ii), observe that the naive Frobenius functor [cf. Proposition 2.1] determines a natural 1-commutative diagram [cf. Proposition 4.4, (ii), (iv)] Ψ C ⏐ ⏐ −→ C birat Ψ birat −→ C ⏐ ⏐ C birat in which the vertical arrows are the natural functor C → C birat of Proposition 4.4, (i); the horizontal arrows are the “naive Frobenius functor” of Proposition 2.1; the upper horizontal arrow is an equivalence of categories [by our assumption that C is of perfect type; Proposition 2.1, (iii)]. Since, moreover, Ψ and any quasi-inverse to Ψ preserve [necessarily co-angular, since C is of isotropic type] pre-steps, it thus follows immediately [cf. the deﬁnition of “C birat ”] that Ψ birat is also an equivalence of categories. But this implies [cf. Proposition 2.1, (iii)] that C birat is of perfect type, as desired. In light of assertion (i), this completes the proof of assertion (ii). Finally, assertion (iii) follows formally from the deﬁnitions [cf. also assertion (i)]; assertion (iv) follows formally Proposition 4.4, (iv) [cf. also assertion (i)]. We are now ready to “reconstruct Φ category-theoretically from C”. Theorem 4.9. (Category-theoreticity of Divisor Monoids) For i = 1, 2, let Φ i be a divisorial monoid on a connected, totally epimorphic category D i ; C i → F Φ i a Frobenioid of rationally standard type; ∼ Ψ : C 1 → C 2 an equivalence of categories. Then there exists an isomorphism of functors ∼ Ψ Φ : Φ 1 → Φ 2 [where we regard, for i = 1, 2, the functor Φ i : D i → Mon as a functor on C i , by restriction via the natural projection functor C i → D i ] lying over Ψ, which is THE GEOMETRY OF FROBENIOIDS I 89 compatible [when the C i are of isotropic, but not of group-like type] with the isomorphism Ψ Prime of Theorem 4.2, (ii). Proof. First, we observe [cf. Theorem 3.4, (i), (ii)] that we may assume without loss of generality that C 1 , C 2 are of isotropic type [cf. Remark 4.5.1], but not of group-like type [since Theorem 4.9 is vacuous if C 1 , C 2 are of group-like type]. Next, I claim that to complete the proof of Theorem 4.9, it suﬃces to show that the right-hand and left-hand isomorphisms of Theorem 4.2, (iii), coincide [cf. Re- mark 4.9.1 below], for all universally Div-Frobenius-trivial objects [e.g., Frobenius- trivial objects — cf. Remark 1.11.1]. Indeed, if the right-hand and left-hand iso- morphisms of Theorem 4.2, (iii), coincide for all universally Div-Frobenius-trivial objects, then it follows immediately from the construction of the isomorphism of functors Ψ Prime in the proof of Theorem 4.2, (ii), that Ψ Prime extends, for A i ∈ Ob(C i bs-iso ), p i ∈ Prime(Φ i (A i )) [where i = 1, 2] such that A 2 = Ψ(A 1 ), p 2 = Ψ Prime (p 1 ), to an isomorphism of monoids ∼ pf Φ 1 (A 1 ) pf p 1 → Φ 2 (A 2 ) p 2 which is functorial in A 1 [regarded as an object of C 1 bs-iso ]. Thus, by allowing the p i to vary, we obtain, for A i ∈ Ob(C i bs-iso ) [where i = 1, 2] such that A 2 = Ψ(A 1 ), an isomorphism of monoids ∼ pf Φ 1 (A 1 ) pf factor → Φ 2 (A 2 ) factor [cf. Deﬁnition 2.4, (i), (c)] which is functorial in A 1 [regarded as an object of C 1 bs-iso ]. Moreover, by applying, say, the ﬁrst equivalence of categories of Deﬁnition 1.3, (iii), (d), to obtain pre-steps φ : A → B of C i with arbitrary prescribed zero divisor and considering primary steps ψ : A → C such that φ = ζ ◦ ψ for some pre-step ζ, one concludes immediately that this subset maps the subset Φ 1 (A 1 ) ⊆ Φ 1 (A 1 ) pf factor [cf. pf Deﬁnition 2.4, (i), (c)] onto the subset Φ 2 (A 2 ) ⊆ Φ 2 (A 2 ) factor , hence determines an isomorphism of monoids ∼ Φ 1 (A 1 ) → Φ 2 (A 2 ) which is functorial in A 1 [regarded as an object of C 1 bs-iso ]. Finally, the functoriality of this isomorphism of monoids with respect to pull-back morphisms follows imme- diately by “pulling back pre-steps”, as in Proposition 1.11, (v). This completes the proof of the claim. To prove that the right-hand and left-hand isomorphisms of Theorem 4.2, (iii), coincide for all universally Div-Frobenius-trivial objects, we reason as follows. First of all, by passing to perfections [cf. Theorem 3.4, (iii)], we may assume without loss of generality that C 1 , C 2 are of perfect type [cf. also Proposition 5.5, (iii), below]. Let A be a universally Div-Frobenius-trivial object of C i [where i = 1, 2]. Since the right-hand and left-hand isomorphisms of Theorem 4.2, (iii), are clearly compatible with pull-back morphisms [cf. Proposition 1.11, (v); the proof of Theorem 4.2, (iii)], and Ψ preserves pull-back morphisms [cf. Theorem 3.4, (iii)], it follows that we may assume without loss of generality that A is strictly rational. Let us refer to pairs 90 SHINICHI MOCHIZUKI of primary steps β : A → B, γ : C → A such that Div(β) = (Φ i (γ)) −1 (Div(γ)) as twin-primary steps. Then, it suﬃces to show, for each p ∈ Prime(Φ 1 (A)), the existence of twin-primary steps with zero divisor in p that are mapped by Ψ to twin-primary steps of C 2 . On the other hand, since A is strictly rational, it follows [cf. Deﬁnition 4.5, (ii)] that there exist, for each p ∈ Prime(Φ i (A)), cartesian commutative diagrams of pre-steps as in Proposition 4.1, (iii), γ γ C −→ D ⏐ ⏐ ⏐ ⏐ γ C −→ D ⏐ ⏐ ⏐ ⏐ γ δ B β −→ A A δ α −→ F def in which α, β are twin-primary with zero divisor in p; the pre-steps ζ = β ◦ γ : C → A, γ : C → A are Div-equivalent [e.g., base-equivalent]. [Indeed, Deﬁnition 4.5, (ii) [cf. also the equivalences of categories of Deﬁnition 1.3, (iii), (d)], guarantees the existence of base-equivalent pre-steps ζ, γ — which may, moreover, be taken to be co-primary [cf. Proposition 4.1, (iii); Deﬁnition 2.4, (i), (c), (d)], by our assumption that C i is of perfect type — such that ζ admits a factorization β ◦ γ, where β is primary with zero divisor that maps via Φ(β) −1 to an element of p, and [again by our assumption that C i is of perfect type] p is not contained in the support of (Φ(ζ)) −1 (Div(γ)).] Conversely, given any pair of cartesian diagrams of pre-steps as in Proposition 4.1, (iii), γ γ C −→ D ⏐ ⏐ ⏐ γ ⏐ C −→ D ⏐ ⏐ ⏐ ⏐ γ δ B β −→ A A δ α −→ F def in which α, β are primary with zero divisor in p; the pre-steps ζ = β ◦ γ : C → A, γ : C → A are Div-equivalent [e.g., base-equivalent], it follows immediately that α, β are twin-primary. On the other hand, since Ψ preserves pre-steps [cf. Theorem 3.4, (ii)], primary steps [cf. Theorem 4.2, (i)], Div-equivalent pairs of base- isomorphisms [cf. Theorem 4.2, (ii); the fact that Φ i is non-dilating], and cartesian diagrams as in Proposition 4.1, (iii) [cf. Proposition 4.1, (iii), or, alternatively, Theorem 4.2, (ii)], we thus conclude that for each p ∈ Prime(Φ 1 (A)), there exist twin-primary steps with zero divisor in p that are mapped by Ψ to twin-primary steps of C 2 . This completes the proof of Theorem 4.9. Remark 4.9.1. One veriﬁes immediately that the Frobenioid of Example 4.3 is not of rational type. Corollary 4.10. (Category-theoreticity of the Birationalization) For i = 1, 2, let Φ i be a divisorial monoid on a connected, totally epimorphic category D i of FSMFF-type; C i → F Φ i a Frobenioid of quasi-isotropic type; ∼ Ψ : C 1 → C 2 THE GEOMETRY OF FROBENIOIDS I 91 an equivalence of categories. Then there exists a 1-unique functor Ψ birat : C 1 birat → C 2 birat that ﬁts into a 1-commutative diagram Ψ C 1 ⏐ ⏐ −→ C 1 birat Ψ birat −→ C 2 ⏐ ⏐ C 2 birat [where the vertical arrows are the natural functors of Proposition 4.4, (i); the hor- izontal arrows are equivalences of categories]. Finally, if D 1 , D 2 are slim, and C 1 , C 2 are of birationally Frobenius-normalized type, then each of the composite functors of this diagram is rigid. Proof. The existence and 1-uniqueness of a 1-commutative diagram as in the statement of Corollary 4.10 follows immediately from the deﬁnition of “C i birat ” [cf. Proposition 4.4, (i)], and the fact that Ψ preserves co-angular pre-steps [cf. Theorem 3.4, (ii)]. The rigidity assertion then follows immediately from Proposition 1.13, (i), by considering base-identity endomorphisms of Frobenius type of Frobenius-trivial objects of C i , under the hypothesis that the C i are birationally Frobenius-normalized [cf., e.g., the proof of the rigidity assertion of Theorem 3.4, (i)]. Corollary 4.11. (Category-theoreticity of the Functor to an Elementary Frobenioid I) For i = 1, 2, let Φ i be a perf-factorial divisorial monoid on a connected, totally epimorphic category D i which is Div-slim [with respect to Φ i ]; C i → F Φ i a Frobenioid of standard type; ∼ Ψ : C 1 → C 2 an equivalence of categories. If C 1 , C 2 are of group-like type, then we also assume that both Ψ and some quasi-inverse to Ψ preserve base-isomorphisms. Then: (i) There exists a 1-unique functor Ψ un-tr : C 1 un-tr → C 2 un-tr that ﬁts into a 1-commutative diagram Ψ istr −→ C 2 istr C 1 istr ⏐ ⏐ ⏐ ⏐ C 1 un-tr Ψ un-tr −→ C 2 un-tr [where the vertical arrows are the natural projection functors; the horizontal arrows are equivalences of categories; Ψ istr is the restriction of Ψ to C 1 istr — cf. Theorem 3.4, (i)]. Moreover, each of the composite functors of this diagram is rigid. (ii) There exists a 1-unique functor Ψ Base : D 1 → D 2 that ﬁts into a 1- commutative diagram Ψ C 1 −→ C 2 ⏐ ⏐ ⏐ ⏐ D 1 Ψ Base −→ D 2 92 SHINICHI MOCHIZUKI [where the vertical arrows are the natural projection functors; the horizontal arrows are equivalences of categories]. Moreover, if D 1 , D 2 are slim, then each of the composite functors of this diagram is rigid. (iii) Suppose further that C 1 , C 2 are of rationally standard type. Then there exists an isomorphism of functors ∼ Ψ Φ : Φ 1 → Φ 2 [where we regard, for i = 1, 2, the functor Φ i : D i → Mon as a functor on D i ] lying over the equivalence of categories Ψ Base of (i), which is compatible [when the C i are of isotropic, but not of group-like type] with the isomorphism Ψ Prime of Theorem 4.2, (ii). In particular, Ψ Base , Ψ Φ induce an equivalence of categories ∼ Ψ F : F Φ 1 → F Φ 2 . (iv) Suppose further that C 1 , C 2 are of rationally standard type. Then there exists a 1-commutative diagram Ψ C 1 ⏐ ⏐ −→ F Φ 1 −→ Ψ F C 2 ⏐ ⏐ F Φ 2 [where the vertical arrows are the functors that deﬁne the Frobenioid structures on C 1 , C 2 ; the horizontal arrows are equivalences of categories]. Moreover, each of the composite functors of this diagram is rigid. Proof. First, we observe [cf. Theorem 3.4, (i)] that we may assume without loss of generality that C 1 , C 2 are of isotropic type [cf. Remark 4.5.1]. Also, if C 1 , C 2 are of group-like type [cf. Theorem 3.4, (ii)], then “Div-slimness” amounts to “slimness”, so assertions (i), (ii) follow from Theorem 3.4, (iv), (v); assertion (iii) is vacuous; assertion (iv) follows from the fact that Ψ preserves Frobenius degrees [cf. Theorem 3.4, (iii), (iv)]. Thus, we may assume without loss of generality that C 1 , C 2 are not of group-like type. Now we consider assertion (i). To show the existence and 1-uniqueness of a 1-commutative diagram as in the statement of assertion (i), it suﬃces to show that Ψ preserves “O × (−)” [cf. the proof of Theorem 3.4, (iv)]. But observe that, for A ∈ Ob(C i ), an element f ∈ O × (A) determines an automorphism [cf. the proof of Proposition 3.3, (i)] φ f ∈ Aut((C i pl-bk ) A → C i ) def that maps to the identity in Aut((D i ) A D → D i ) [where A D = Base(A) — cf. ∼ the equivalence of categories (C i pl-bk ) A → (D i ) A D of Deﬁnition 1.3, (i), (c)], hence also to the identity in Aut((D i ) A D → Mon) [i.e., via composition with Φ i ]. Since D i is Div-slim, it thus follows that the elements of O × (A) ⊆ Aut C i (A) may be characterized as the automorphisms of A that arise from automorphisms φ ∈ Aut((C i pl-bk ) A → C i ) THE GEOMETRY OF FROBENIOIDS I 93 such that every automorphism [of an object of C i ] induced by φ is a Div-identity automorphism. Thus, since Ψ preserves pull-back morphisms [cf. Theorem 3.4, (iii)] and Div-identity automorphisms [cf. Theorem 4.2, (i); our assumption that the Φ i are perf-factorial], we thus conclude that Ψ preserves “O × (−)”, as desired. This completes the proof of the existence and 1-uniqueness of a 1-commutative diagram as in the statement of assertion (i). The rigidity assertion in the statement of assertion (i) follows by observing that if α ∈ Aut(C i → C i un-tr ), then every automorphism [of an object of C i un-tr ] induced by α is a Div-identity automorphism. [Indeed, this follows by applying the functoriality of α to [co-angular] pre-steps, in light of the second equivalence of categories of Deﬁnition 1.3, (iii), (d).] In particular, it follows that if A ∈ Ob(C i ), def A D = Base(A), then the element ∼ α A ∈ Aut((C i pl-bk ) A → D i ) → Aut((D i ) A D → D i ) determined by α maps [under composition with Φ i : D i → Mon] to the identity element of Aut((D i ) A D → Mon). Thus, since D i is Div-slim, it follows that every automorphism [of an object of C i un-tr ] induced by α is a base-identity automor- phism, hence trivial [since C i un-tr is of unit-trivial type]. This completes the proof of assertion (i). Next, we consider assertion (ii). First, let us observe that we obtain a 1- commutative diagram Ψ −→ C 2 C 1 ⏐ ⏐ ⏐ ⏐ C 1 birat Ψ birat −→ C 2 birat [cf. Corollary 4.10]. Since, moreover, the base-isomorphisms of C i birat are precisely the morphisms of C i birat which are abstractly equivalent to morphisms that arise from base-isomorphisms of C i [cf. Proposition 4.4, (iv)], it follows that Ψ birat pre- serves base-isomorphisms, hence also pull-back morphisms [cf. Proposition 1.7, (ii)]. Thus, since D i is Div-slim, the base-identity endomorphisms of A ∈ Ob(C i birat ) may be characterized as the endomorphisms of A that arise from endomorphisms → C i birat ) φ ∈ End((C i birat ) pl-bk A such that every endomorphism [of an object of C i birat ] induced by φ projects to an automorphism of D i that is mapped by Φ i to an identity automorphism. Since, by Theorem 4.2, (ii) [cf. also the fact that the Φ i are perf-factorial and non- dilating], it follows immediately from the deﬁnition of C i birat that Ψ birat preserves those endomorphisms [of an object of C i birat ] that project to an automorphism of D i that is mapped by Φ i to an identity automorphism, we thus conclude that Ψ birat preserves the base-identity endomorphisms [hence, in particular, that Ψ birat preserves “O × (−)”]. Thus, we obtain a 1-commutative diagram C 1 birat ⏐ ⏐ (C 1 birat ) un-tr Ψ birat −→ (Ψ birat ) un-tr −→ C 2 birat ⏐ ⏐ (C 2 birat ) un-tr 94 SHINICHI MOCHIZUKI [where the vertical arrows are the natural functors; the horizontal arrows are equiv- alences of categories]. Since, moreover, the Frobenioids (C i birat ) un-tr are of isotropic, unit-trivial, and group-like type, we thus conclude that we obtain a 1-commutative diagram (Ψ birat ) un-tr −→ (C 2 birat ) un-tr (C 1 birat ) un-tr ⏐ ⏐ ⏐ ⏐ D 1 Ψ Base −→ D 2 [cf. Proposition 3.11, (iii)]. Thus, by composing diagrams, we obtain a 1-commutative diagram as in the statement of assertion (ii), which is easily veriﬁed to be 1-unique. Finally, the rigidity assertion in the statement of assertion (ii) follows from Propo- sition 1.13, (i). This completes the proof of assertion (ii). Next, we observe that assertion (iii) follows formally from assertion (ii); Theo- rem 4.9 [cf. also Deﬁnition 1.3, (i), (a), (b); the technique of using the equivalence ∼ of categories “D ∗ → D” applied in Proposition 2.2, (ii)]. Finally, we consider asser- tion (iv). In light of the structure of an elementary Frobenioid [cf. Deﬁnition 1.1, (iii)], the existence of a 1-commutative diagram as in the statement of assertion (iv) now follows simply by concatenating assertions (ii), (iii), with the fact that Ψ preserves Frobenius degrees [cf. Theorem 3.4, (iii)]. Finally, the rigidity assertion follows via the same argument as was applied to prove the rigidity assertion that appears in the statement of assertion (i). This completes the proof of assertion (iv). Remark 4.11.1. Note that since “slim always implies Div-slim”, it follows that, at least when the divisorial monoids involved are perf-factorial, Corollary 4.11, (ii), constitutes a substantial strengthening of Theorem 3.4, (v). Remark 4.11.2. Observe that in Example 3.9, since the subgroup U of G = Aut(D A → D) [where A ∈ Ob(D)] acts trivially on V × W , it follows that D fails to be Div-slim, so the non-preservation of units that occurs in this example does not contradict Corollary 4.11, (i), (ii). In a similar vein, in Example 3.10, since G = Aut(D A → D) [where A ∈ Ob(D)] acts trivially on Z ≥0 , it follows that D fails to be Div-slim, so the non-preservation of base-identity endomorphisms that occurs in this example does not contradict Corollary 4.11, (ii). Corollary 4.12. (Category-theoreticity of the Functor to an Elementary Frobenioid II) For i = 1, 2, let Φ i be a divisorial monoid on a connected, totally epimorphic category D i which is Frobenius-slim; C i → F Φ i a Frobenioid of rationally standard type; 0 D i the monoid on D i that assigns to every object of D i the monoid with one element; ∼ Ψ : C 1 → C 2 an equivalence of categories. If C 1 , C 2 are of group-like type, then we also assume that both Ψ and some quasi-inverse to Ψ preserve base-isomorphisms. Then THE GEOMETRY OF FROBENIOIDS I 95 there exists a 1-unique functor Ψ 0 : F 0 D 1 → F 0 D 2 that ﬁts into a 1-commutative diagram Ψ −→ C 2 C 1 ⏐ ⏐ ⏐ ⏐ F 0 D 1 Ψ 0 −→ F 0 D 2 [where the vertical arrows are the natural projection functors, determined by the Frobenius degree and the projection to D i [cf. Proposition 4.4, (i)]; the horizontal arrows are equivalences of categories]. Moreover, if D 1 , D 2 are slim, then each of the composite functors of this diagram is rigid. Proof. First, we observe [cf. Theorem 3.4, (i)] that we may assume without loss of generality that C 1 , C 2 are of isotropic type [cf. Remark 4.5.1]. Now the natural projection functors C i → F 0 D i may be identiﬁed with the natural functors C i → C i birat → (C i birat ) un-tr [cf. Proposition 3.11, (i)]. In particular, if C 1 , C 2 are ∼ of group-like type [cf. Theorem 3.4, (ii)], then [since C i → C i birat ] Corollary 4.12 follows from Theorem 3.4, (iv). Thus, we may assume without loss of generality that C 1 , C 2 are not of group-like type. Then Corollary 4.12 follows by applying Corollary 4.10 to pass from C i to C i birat [where we note that, by Proposition 4.4, (iv), and Theorem 3.4, (iii), it follows that the resulting equivalence of categories Ψ birat preserves base-isomorphisms], followed by Theorem 3.4, (iv) [where we note that, by Proposition 4.8, (iii), C i birat is of standard type], which allows us to pass from C i birat to (C i birat ) un-tr , as desired. Finally, the rigidity assertion follows from Proposition 1.13, (i). This completes the proof of Corollary 4.12. Remark 4.12.1. One veriﬁes immediately that if one takes the group G of Exam- ple 3.10 to be residually ﬁnite, then the Frobenioid of Example 3.10 is of rationally standard and unit-trivial type [but not of group-like type] over a Frobenius-slim base category [which is not Div-slim — cf. Remark 4.11.2]. In particular, one may apply Corollary 4.12 to the self-equivalence of categories of Example 3.10. On the other hand, since this self-equivalence fails to preserve base-identity endomor- phisms of Frobenius type, it follows that it is not possible to replace the “F 0 D i ” in the diagram of Corollary 4.12 by “D i ”. 96 SHINICHI MOCHIZUKI Section 5: Model Frobenioids In the present §5, we study the extent to which an arbitrary Frobenioid of isotropic type may be constructed explicitly as a “model Frobenioid”. This study of “model Frobenioids” will be of use in the consideration of the concrete examples of Frobenioids that we discuss in §6 below. In the following discussion, we maintain the notation of §1, §2, §3, §4. In particular, we assume that we have been given a divisorial monoid Φ on a connected, totally epimorphic category D and a Frobenioid C → F Φ . Theorem 5.1. (Divisorial Descriptions) Suppose that the Frobenioid C is def of isotropic type. Let A, A ∈ Ob(C) be Frobenius-trivial; A D = Base(A) ∈ def Ob(D); A D = Base(A ) ∈ Ob(D); D isom ⊆ D the subcategory determined by the def isom isomorphisms of D; D D = (D isom ) D [for D ∈ Ob(D) = Ob(D isom )]. Write def Pic Φ (A) = Φ gp (A)/Φ birat (A) [cf. Proposition 4.4, (iii)] and Pic C (A) for the set of isomorphism classes of C × D isom D A [where the ﬁber product category is taken with respect to the natural functors D isom C → D, D A → D — cf. §0]. Then: D (i) The assignment that maps a pair of pre-steps (φ : B → A, ψ : B → C) to the object isom (C, Base(φ) ◦ Base(ψ) −1 ) ∈ Ob(C × D D A ) D on the one hand and to the element Φ(φ) −1 (Div(ψ) − Div(φ)) ∈ Φ gp (A) ∼ on the other hand determines a bijection Pic Φ (A) → Pic C (A). Moreover, if (C, ζ : ∼ def isom C D → A D ) ∈ Ob(C × D D A ) [where C ∈ Ob(C), C D = Base(C)] corresponds, D via this bijection, to an element γ ∈ Pic Φ (A), and κ : C → C is a morphism of isom Frobenius type, then (C , ζ ◦ Base(κ) −1 ) ∈ Ob(C × D D A ) corresponds to the D element deg Fr (κ) · γ ∈ Pic Φ (A). (ii) If ∼ isom ); (B, λ : B D → A D ) ∈ Ob(C × D D A D def ∼ isom (B , λ : B D → A D ) ∈ Ob(C × D D A ) def [where B D = Base(B); B D = Base(B )], then there exists a morphism φ : B → B D THE GEOMETRY OF FROBENIOIDS I 97 in C of Frobenius degree d such that Base(φ) = (λ ) −1 ◦ θ ◦ λ, where θ : A D → A D is a morphism of D, and Div(φ) = z ∈ Φ(B) if and only if the classes β ∈ Pic Φ (A), β ∈ Pic Φ (A ) determined by B, B , respectively, via the bijection of (i) satisfy the following relation: d · β + z| A D = (Φ(θ))(β ) ∈ Pic Φ (A) [where, by abuse of notation, we denote by z| A D the image of Φ(λ) −1 (z) ∈ Φ(A) in Pic Φ (A)]. Moreover, if such a morphism exists, then its unit-equivalence class [i.e., its image in C un-tr , or, equivalently, F Φ — cf. Proposition 3.3, (iv)] is unique. (iii) The subcategory C Fr-tr ⊆ C determined by the Frobenius-trivial objects and isometric morphisms is a Frobenioid of isotropic, group-like, base-trivial, and Aut-ample type. In particular, the isomorphism class of a Frobenius-trivial object of C is completely determined by the isomorphism class of its projection to D; all Frobenius-trivial objects of C are Aut-ample. (iv) Suppose that C is of unit-trivial type. Then any skeletal subcategory P ⊆ (C Fr-tr ) pl-bk determines a base-section of C; any base-section of C admits an associated Frobenius-section F . Moreover, C is of model type. Proof. First, we consider assertion (i). Let us refer to a(n) [ordered] pair of pre- steps as an A-pair if the ﬁrst pre-step has codomain A, and the second pre-step has the same domain as the ﬁrst; let us say that two A-pairs (φ : B → A, ψ : B → C); ∼ (φ : B → A, ψ : B → C ) are isomorphic if there exist isomorphisms ι B : B → B , ∼ ι C : C → C such that φ ◦ ι B = φ, ψ ◦ ι B = ι C ◦ ψ. Then observe that by the equivalences of categories of Deﬁnition 1.3, (iii), (d), it follows that the assignment (φ : B → A, ψ : B → C) → (Φ(φ) −1 (Div(φ)), Φ(φ) −1 (Div(ψ))) ∈ Φ(A) × Φ(A) determines a bijection from the set of isomorphism classes of A-pairs onto Φ(A) × Φ(A); in particular, we obtain a map Φ(A) × Φ(A) → Pic C (A). Moreover, relative to this bijection, replacing an element (x, y) ∈ Φ(A) × Φ(A) by an element (x + z, y + z) ∈ Φ(A) × Φ(A) [where z ∈ Φ(A)] corresponds to replacing (φ : B → A, ψ : B → C) by (φ ◦ δ, ψ ◦ δ), for some pre-step δ; in particular, such replacements do not aﬀect the element of Pic C (A) determined by the A-pair. Now I claim that the map Φ(A) × Φ(A) → Pic C (A) of the above discussion factors through Pic Φ (A). Indeed, suppose that (x, y) ∈ Φ(A) × Φ(A), (x , y ) ∈ Φ(A) × Φ(A) map to the same element of Pic Φ (A). Then, by the deﬁnition of “Φ birat (A)” [cf. the statements and proofs of Proposition 4.4, (i), (iii)], it follows that there exists a pair of base-equivalent pre-steps δ 1 , δ 2 : D → A such that Φ(δ 1 ) −1 (Div(δ 1 )) + x + y + z = Φ(δ 2 ) −1 (Div(δ 2 )) + x + y + z for some z ∈ Φ(A) [cf. also the deﬁnition of “gp” in §0]; thus, by replacing δ 1 , δ 2 by the composite of δ 1 , δ 2 with an appropriate pre-step [cf. Deﬁnition 1.3, (iii), (d)], we may assume that Φ(δ 1 ) −1 (Div(δ 1 )) = x + y + z ; Φ(δ 2 ) −1 (Div(δ 2 )) = x + y + z 98 SHINICHI MOCHIZUKI for some z ∈ Φ(A) [for instance, one natural choice for z is Φ(δ 1 ) −1 (Div(δ 1 )) + x + y + z = Φ(δ 2 ) −1 (Div(δ 2 )) + x + y + z]; by replacing (x, y) by (x + z , y + z ) [cf. discussion of the the preceding paragraph], it follows that we may assume, without loss of generality, that z = 0. Next, by applying the ﬁrst equivalence of categories of Deﬁnition 1.3, (iii), (d), we observe that there exists a pre-step δ † : D → D † such that Div(δ † ) = Φ(δ i )(x + x + y + y ), where i = 1, 2 [and we note that Φ(δ i ) is independent of i, since δ 1 , δ 2 are base-equivalent]. Thus, [again by Deﬁnition 1.3, (iii), (d)] we conclude that there exist base-equivalent pre-steps δ 1 A , δ 2 A : A → D † such that δ † = δ 2 A ◦ δ 1 = δ 1 A ◦ δ 2 . In particular, we have Div(δ 1 A ) = x + y , Div(δ 2 A ) = x + y. Let : E → A be a pre-step with Φ( ) −1 (Div( )) = x + x [cf. Deﬁnition 1.3, (iii), (d)]; (φ : B → A, ψ : B → C) an A-pair that corresponds to (x, y); (φ : B → A, ψ : B → C ) an A-pair that corresponds to (x , y ). Then since x, x ≤ x + x , it follows [cf. Deﬁnition 1.3, (iii), (d)] that there exist factorizations = φ ◦ η, = φ ◦ η , where η : E → B, η : E → B are pre-steps. Moreover, by applying the the second equivalence of categories of Deﬁnition 1.3, (iii), (d), to D † , we conclude from the existence of the composites of : E → A with δ 1 A , δ 2 A : A → D † that there exists a pre-step F : F → D † and a pair of base-equivalent pre-steps δ 1 E , δ 2 E : E → F such that the following relations hold: F ◦ δ 1 E = δ 1 A ◦ ; F Div(δ 1 E ) = (Φ( ))(x + y ); ◦ δ 2 E = δ 2 A ◦ Div(δ 2 E ) = (Φ( ))(x + y) [so Φ( F ) −1 (Div( F )) = Φ(δ i A ) −1 (x + x ), for i = 1, 2]. On the other hand, since Div(ψ ◦ η) = (Φ( ))(x + y) = Div(δ 2 E ), Div(ψ ◦ η ) = (Φ( ))(x + y ) = Div(δ 1 E ), we thus conclude [cf. Deﬁnition 1.3, (iii), (d), applied to the pairs of pre-steps (ψ ◦ η : E → C, δ 2 E : E → F ) and (ψ ◦ η : E → C , δ 1 E : E → F ) emanating from E] ∼ that there exists an isomorphism ι : C → C such that Base(ψ ◦η ) = Base(ι◦ψ ◦η), Base(ι ◦ ψ) ◦ Base(φ) −1 = Base(ψ ) ◦ Base(φ ) −1 . That is to say, we have a [not necessarily commutative!] diagram of pre-steps η E −→ ⏐ ⏐ η φ A ←− B φ B −→ A ⏐ ⏐ ι◦ψ ψ −→ C whose projection to D is a commutative diagram of isomorphisms that is compat- ible with identiﬁcation of the two copies of A D . In particular, we conclude that (C, Base(φ) ◦ Base(ψ) −1 ), (C , Base(φ ) ◦ Base(ψ ) −1 ) determine the same element of Pic C (A). This completes the proof of the claim. Thus, we obtain a map Pic Φ (A) → Pic C (A). It follows immediately from Deﬁnition 1.3, (i), (b), that this map is a surjection. To show that this map is injective, it suﬃces to consider (x, y) ∈ Φ(A) × Φ(A), (x , y ) ∈ Φ(A) × Φ(A) that map to the same element of Pic C (A). Let (φ : B → A, ψ : B → C) be an A-pair that corresponds to (x, y); (φ : B → A, ψ : B → C) an A-pair that THE GEOMETRY OF FROBENIOIDS I 99 corresponds to (x , y ). By our assumption that (x, y) and (x , y ) map to the same element of Pic C (A), it follows that we may assume that Base(φ ) ◦ Base(ψ ) −1 = Base(φ) ◦ Base(ψ) −1 . Thus, by applying Deﬁnition 1.3, (iii), (d), we obtain a [not necessarily commutative!] diagram of pre-steps η φ E −→ B −→ A ⏐ ⏐ ⏐ ⏐ ψ η A φ ←− B ψ −→ C such that φ ◦ η = φ ◦ η , and whose projection to D is a commutative diagram of isomorphisms that is compatible with identiﬁcation of the two copies of A D . Thus, it follows that ψ ◦ η, ψ ◦ η : E → C are base-equivalent, hence determine an element of Φ birat (C), which may be transported via ψ, φ [or, equivalently, ψ , φ ] to an element of Φ birat (A) ⊆ Φ gp (A) which [cf. the discussion of the preceding paragraph] is easily veriﬁed to be x +y −x −y ∈ Φ gp (A). This completes the proof of the injectivity, hence also of the bijectivity of the map Pic Φ (A) → Pic C (A). Also, the portion of assertion (i) concerning morphisms of Frobenius type follows easily by considering commutative diagrams as in Proposition 1.10, (i). This completes the proof of assertion (i). Now assertion (ii) follows formally from assertion (i) [cf. also Remark 1.1.1; the factorization of Deﬁnition 1.3, (iv), (a); the faithfulness portion of Proposition 3.3, (iv)]. Next, we consider assertion (iii). First, let us observe that by assertion (i), any ∼ isom isomorphism α D : A D → A D determines an object (A , α) ∈ Ob(C × D D A ) which D [in light of the fact that A is Frobenius-trivial, hence admits base-identity endo- morphisms of Frobenius type of arbitrary prescribed Frobenius degree] corresponds [via the bijection of assertion (i)] to an element ξ ∈ Pic Φ (A) such that d · ξ = ξ, for all d ∈ N ≥1 . Thus, taking d = 2 implies that ξ = 0, i.e., [cf. the deﬁnition of ∼ Pic C (A)] that there exists an isomorphism α : A → A such that α D = Base(α). In particular, we conclude that base-isomorphic Frobenius-trivial objects of C are, in fact, isomorphic, and that all Frobenius-trivial objects of C are Aut-ample. In light of these observations, it follows immediately that C Fr-tr satisﬁes the conditions of Deﬁnition 1.3, i.e., that C Fr-tr is a Frobenioid [of isotropic, group-like, base-trivial, and Aut-ample type]. This completes the proof of assertion (iii). Finally, we consider assertion (iv). First, we observe that since C is of unit- trivial type, it follows immediately [cf., e.g., Proposition 3.3, (iii), (iv)] that given any two objects A, B ∈ Ob(C), a pull-back morphism A → B (respectively, base- identity endomorphism of Frobenius type of A) is uniquely determined by its projec- tion to D (respectively, by its Frobenius degree). Moreover, by assertion (iii), it fol- lows immediately that if A, B ∈ Ob(C Fr-tr ), then any morphism Base(A) → Base(B) [in D] lifts to a pull-back morphism of C Fr-tr . Thus, we conclude that the natural projection functor (C Fr-tr ) pl-bk → D is an equivalence of categories, hence that any skeletal subcategory P ⊆ (C Fr-tr ) pl-bk determines a base-section of C, and that any base-section of C admits an associated 100 SHINICHI MOCHIZUKI Frobenius-section. Moreover, since C is of unit-trivial type, it follows immediately from the structure of an elementary Frobenioid [cf. the description of the kernel in Proposition 4.4, (iii)] that C is of birationally Frobenius-normalized type, hence also of model type, as desired. This completes the proof of assertion (iv). The explicit descriptions of Theorem 5.1, (i), (ii), motivate the following con- struction/result. Theorem 5.2. (Model Frobenioids) Let Φ : D → Mon be a divisorial monoid on D; B : D → Mon a group-like monoid on D; Div B : B → Φ gp a homomorphism of monoids on D. Denote the group-like monoid determined by the image of Div B by Φ birat ⊆ Φ gp . Then: (i) A well-deﬁned category C may be constructed in the following fashion. The objects of C are pairs of the form (A D , α) def def def where A D ∈ Ob(D), α ∈ Φ(A D ) gp ; set Base(A) = A D , Φ(A) = Φ(A D ), B(A) = B(A D ). A morphism def def φ : A = (A D , α) → B = (B D , β) [where A D , B D ∈ Ob(D), α ∈ Φ(A) gp , β ∈ Φ(B) gp ] is deﬁned to be a collection of data as follows: (a) an element deg Fr (φ) ∈ N ≥1 , which we shall refer to as the Frobenius degree of φ; (b) a morphism Base(φ) : A D → B D , which we shall refer to as the projection to D to φ; (c) an element Div(φ) ∈ Φ(A), which we shall refer to as the zero divisor of φ; (d) an element u φ ∈ B(A) whose image Div B (u φ ) ∈ Φ(A) gp satisﬁes the relation deg Fr (φ) · α + Div(φ) = (Φ(Base(φ)))(β) + Div B (u φ ) in Φ(A) gp . The composite ψ ◦ φ of two morphisms φ = (deg Fr (φ), Base(φ), Div(φ), u φ ) : A → B ψ = (deg Fr (ψ), Base(ψ), Div(ψ), u ψ ) : B → C is deﬁned as follows: ψ ◦ φ = deg Fr (ψ) · deg Fr (φ), Base(ψ) ◦ Base(φ), def (Φ(Base(φ)))(Div(ψ)) + deg Fr (ψ) · Div(φ), (B(Base(φ)))(u ψ ) + deg Fr (ψ) · u φ [cf. Remark 1.1.1]. Moreover, the Frobenius degree, projection to D, and zero divisor determine a functor C → F Φ . THE GEOMETRY OF FROBENIOIDS I 101 (ii) The category C is a Frobenioid [with respect to the functor C → F Φ ] of isotropic and model — hence, in particular, birationally Frobenius-normalized — type. We shall refer to C as the model Frobenioid deﬁned by the divisor monoid Φ and the rational function monoid B [which we regard as equipped with the homomorphism Div B : B → Φ gp ]. Moreover, there is a natural isomor- phism of functors between the functor “O × (−)” on D associated to the Frobenioid C birat [cf. Propositions 2.2, (ii), (iii); 4.4, (ii)] and the functor B; this isomorphism is compatible with the homomorphisms O × (−) → Φ gp [cf. Proposition 4.4, (iii)], Div B : B → Φ gp . (iii) C is of standard type if and only if the following conditions are satisﬁed: (a) if Φ is the zero monoid, then C admits a Frobenius-compact object; (b) D is of FSMFF-type; (c) Φ is non-dilating. C is of rationally standard type if and only if the following conditions are satisﬁed: (a) C is of rational and standard type; (b) (C un-tr ) birat admits a Frobenius-compact object. (iv) Suppose that Φ = Φ; B is the rational function monoid on D associated to the Frobenioid C [cf. Proposition 4.4, (ii)]; Div B : B → Φ gp is the natural homomorphism O × (−) → Φ gp = Φ gp [cf. Proposition 4.4, (iii)]; C is of model type. Then there exists an equivalence of categories ∼ C →C that is 1-compatible with the functors C → F Φ , C → F Φ . Proof. Assertions (i), (ii) follow via a routine veriﬁcation [which, in the case of assertion (ii), is reminiscent of the veriﬁcation that “elementary Frobenioids are Frobenioids” in Proposition 1.5, (i)]; in light of assertion (ii), assertion (iii) follows formally from the deﬁnitions [cf. Deﬁnitions 3.1, (i); 4.5, (iii)]. Here, we observe that the objects A = (A D , α) such that α = 0 are Frobenius-trivial, and that these objects, together with the morphisms φ = (deg Fr (φ), Base(φ), Div(φ), u φ ) : A → B such that Div(φ) = 0, u φ = 1 [i.e., u φ is the identity element of B(A)], determine a base-Frobenius pair of C. Finally, we consider assertion (iv). We may assume without loss of generality that C, hence also C Fr-tr , is a skeleton. Let (P, F ) be a base-Frobenius pair of C [cf. our assumption that C is of model type]. Thus, P may be regarded as a subcategory of C Fr-tr . If C ∈ Ob(C), then let us refer to a(n)[ordered] pair of pre-steps in C (B → A, A → C) such that A ∈ Ob(P) as an F P-path for C. Write C for the category C whose objects are objects of C equipped with an F P-path, and whose morphisms are the morphisms between the objects regarded as objects of C. Thus, we have a natural functor C → C [obtained by forgetting the F P-paths], 102 SHINICHI MOCHIZUKI which is manifestly an equivalence of categories. Thus, it suﬃces to construct an ∼ equivalence of categories C → C that is compatible with the functors C → C → F Φ , C → F Φ . Next, observe that we may apply Remark 2.7.2 to C Fr-tr [which is of base-trivial type, by Theorem 5.1, (iii)] to conclude that every morphism φ of C Fr-tr admits a unique factorization φ = φ P ◦ φ O × ◦ φ F in C Fr-tr , where φ P is P-distinguished; φ O × is a base-identity automorphism; φ F is F -distinguished. Let us write E ⊆ C birat for the full subcategory determined by the image of the objects in P. Then observe ∼ that the category E is also a skeleton; that the Frobenioid E → C birat is also of isotropic and base-trivial type [cf. Proposition 4.8, (i); Theorem 5.1, (iii)]; and that (P, F ) determine a base-Frobenius pair of E. Thus, we may apply Remark 2.7.2 to E to conclude that every morphism ψ of E admits a unique factorization ψ = ψ P ◦ ψ O × ◦ ψ F in E, where ψ P is P-distinguished; ψ O × is a base-identity automorphism; ψ F is F -distinguished. Now observe that to every object C ∈ Ob(C) equipped with an F P-path (ζ A : B → A, ζ C : B → C), we may associate an object (Base(A), Φ(ζ A ) −1 (Div(ζ C ) − Div(ζ A )) ∈ Φ gp (A)) of C [cf. Theorem 5.1, (i)]. If C ∈ Ob(C) is equipped with an F P-path (ζ A : B → A , ζ C : B → C ), then we may associate to any morphism φ : C → C a morphism deg Fr (φ), Base(ζ A ) ◦ Base(ζ C ) −1 ◦ Base(φ) ◦ Base(ζ C ) ◦ Base(ζ A ) −1 : A → A , (Φ(ζ A ) −1 ◦ Φ(ζ C ))(Div(φ)) ∈ Φ(A), birat birat −1 birat birat −1 {ζ A ◦ (ζ C ) ◦ φ birat ◦ ζ C ◦ (ζ A ) } O × ∈ O × (A birat ) [where the superscript “birat’s” denote the images of the respective objects and morphisms of C in C birat ] of C. Now in light of the fact that C is of model — hence, in particular, birationally Frobenius-normalized — type, it is a routine exercise to verify that these assignments determine a functor C → C that is compatible with the functors C → C → F Φ , C → F Φ . Indeed, this is immediate for the ﬁrst three entries of the data that deﬁne a morphism of C; for the ﬁnal entry, it follows from the existence of the unique factorizations of morphisms of E discussed above. Note that these factorizations also imply that this functor THE GEOMETRY OF FROBENIOIDS I 103 C → C is faithful. Moreover, this functor C → C is manifestly essentially surjective [cf. Theorem 5.1, (i)] and full [cf. Theorem 5.1, (ii)], hence an equivalence of categories, as desired. This completes the proof of assertion (iv). Remark 5.2.1. It follows formally from Theorem 5.2, (ii), (iv), that the Frobe- nioid “C” of Example 4.6 constitutes an example of a Frobenioid of isotropic, stan- dard, and [strictly] rational type, which is not of group-like or model type. Proposition 5.3. (Realiﬁcations of Frobenioids) Suppose that Φ is perf- factorial. Then we shall refer to as the realiﬁcation C rlf of the Frobenioid C the model Frobenioid [cf. Theorem 5.2, (ii)] associated to the divisor monoid Φ rlf [i.e., the “realiﬁcation” of Deﬁnition 2.4, (i)] and the rational function monoid R · Φ birat ⊆ (Φ rlf ) gp [i.e., for A D ∈ Ob(D), (R · Φ birat )(A D ) is the R-vector subspace of (Φ rlf ) gp (A D ) generated by Φ birat (A D )]. Moreover, the Frobenioid C un-tr (respec- tively, (C un-tr ) pf ) is of model type and may be obtained as the model Frobenioid associated to the divisor monoid Φ (respectively, Φ pf ) and the rational function monoid Φ birat (respectively, Q · Φ birat = Φ birat ⊗ Z Q = (Φ birat ) pf ). In particular, if C is of Frobenius-isotropic type, then there is a natural 1-commutative diagram of functors C −→ C istr ⏐ ⏐ −→ C pf ⏐ ⏐ C un-tr −→ (C un-tr ) pf −→ C rlf [where the functor C → C istr is the isotropiﬁcation functor of Proposition 1.9, (v); the remaining functors are the functors that arise naturally from the construction of the “unit-trivialization”, “perfection”, and “realiﬁcation”]. Proof. Since Frobenioids of unit-trivial type are always of model type [cf. The- orem 5.1, (iv)], the various assertions in the statement of Proposition 5.3 follow immediately from the deﬁnitions and Theorem 5.2, (ii), (iv). Corollary 5.4. (Category-theoreticity of the Realiﬁcation) For i = 1, 2, let Φ i be a perf-factorial divisorial monoid on a connected, totally epimorphic category D i which is Div-slim [with respect to Φ i ]; C i → F Φ i a Frobenioid of rationally standard type; ∼ Ψ : C 1 → C 2 an equivalence of categories. If C 1 , C 2 are of group-like type, then we also assume that both Ψ and some quasi-inverse to Ψ preserve base-isomorphisms. Then 104 SHINICHI MOCHIZUKI there exists a 1-unique functor Ψ rlf : C 1 rlf → C 2 rlf that ﬁts into a 1-commutative diagram Ψ C 1 −→ C 2 ⏐ ⏐ ⏐ ⏐ C 1 rlf Ψ rlf −→ C 2 rlf [where the vertical arrows are the natural functors of Proposition 5.3; the horizontal arrows are equivalences of categories]. Moreover, each of the composite functors of this diagram is rigid. Finally, the formation of Ψ rlf from Ψ is 1-compatible with the 1-commutative diagram of Proposition 5.3 [involving perfections, unit- trivializations, etc.]. Proof. In light of the deﬁnition of the realiﬁcation [cf. Proposition 5.3], Corollary 5.4 follows immediately from Corollaries 4.10; 4.11, (iii), (iv). [Here, we note that the rigidity assertion of Corollary 5.4 follows by a similar argument applied to prove the rigidity assertion in Corollary 4.11, (i), (iv).] Before continuing, we note the following [portions of which were in fact applied in the proofs of Theorems 4.2, 4.9]. Proposition 5.5. (Perfection, Unit-trivialization and Realiﬁcation of Types) Suppose that C is of Frobenius-isotropic and Frobenius-normalized type. Then: (i) If A ∈ Ob(C istr ) maps to an object A pf ∈ Ob(C pf ), then the natural functor ∼ C → C pf determines a natural isomorphism O (A) pf → O (A pf ). (ii) There is a natural equivalence of categories [compatible with the func- tors to the respective elementary Frobenioids] between (C pf ) un-tr and (C un-tr ) pf and between (C pf ) birat and (C birat ) pf . (iii) If C is of standard (respectively, rationally standard; model) type, then so is C pf . Moreover, C un-tr , C rlf are always of model type. Finally, suppose further that C is not of group-like type. Then if C is of standard (respectively, rationally standard) type, then so are C un-tr , C rlf . (iv) If C is the model Frobenioid associated to data Φ, B, Div B : B → Φ gp [cf. Theorem 5.2, (ii)], then there is a natural equivalence of categories [compatible with the functors to the respective elementary Frobenioids] between C pf (respectively, C un-tr ; C rlf ) and the model Frobenioid associated to the data Φ pf , B pf , B pf → (Φ gp ) pf (respectively, Φ, Φ birat , Φ birat → Φ gp ; Φ rlf , R · Φ birat , R · Φ birat → (Φ rlf ) gp ). Proof. Assertion (i) follows immediately for Frobenius-trivial A by considering base-identity endomorphisms of Frobenius type of A and applying the hypothesis that C is of Frobenius-normalized type; the case of arbitrary A then follows by con- sidering “pairs of pre-steps” as in Theorem 5.1, (i) [cf. also Deﬁnition 1.3, (iii), THE GEOMETRY OF FROBENIOIDS I 105 (c)]. Next, we consider assertion (ii). One checks immediately that [in light of our hypothesis that C is of Frobenius-isotropic type] we may assume without loss of gen- erality that C is of isotropic type. Then it follows immediately from the deﬁnition of the perfection [cf. Deﬁnition 3.1, (iii)] that it suﬃces to obtain natural bijections be- tween the respective sets of morphisms between the images of two given objects of C in (C pf ) un-tr , (C un-tr ) pf (respectively, (C pf ) birat ), (C birat ) pf ). But this follows immedi- ately from the deﬁnitions, together with Proposition 3.2, (ii), applied to “pre-steps” and “units” [i.e., base-identity automorphisms]. Next, we consider assertion (iii). First, we observe that C un-tr , C rlf are of model type [cf. Theorem 5.1, (iv); Proposi- tion 5.3; Theorem 5.2, (ii)], hence of isotropic and birationally Frobenius-normalized type [cf. Deﬁnitions 2.7, (iii); 4.5, (i)]. Next, let us observe that by assertion (ii), we ∼ ∼ have natural equivalences ((C un-tr ) birat ) pf → ((C pf ) un-tr ) birat , (C birat ) pf → (C pf ) birat ; moreover, since C un-tr is of birationally Frobenius-normalized type, it follows that (C un-tr ) birat is of Frobenius-normalized type, so assertion (i) may be applied to (C un-tr ) birat . In light of these observations, assertion (iii) for C pf follows immedi- ately from the deﬁnitions [cf. also Proposition 3.2, (ii), (iii)] by observing that C pf is of isotropic type, and that by assertion (i), if C ∗ is C or (C un-tr ) birat [or C birat , when C is of birationally Frobenius-normalized type], and A ∈ Ob((C ∗ ) istr ), then O (−) of the image of A in (C ∗ ) pf is the perfection of O (A). Now suppose that C, hence also C un-tr , C rlf , are not of group-like type. Since (C un-tr ) birat admits a Frobenius- compact object, the same is true for (C rlf ) birat . Also, we observe that the pull-back morphisms of C un-tr , C rlf are precisely the linear isometries [cf. Proposition 1.4, (ii)]. In light of these observations, it follows immediately from the deﬁnitions that if C is of standard (respectively, rationally standard) type, then so are C un-tr , C rlf . Finally, assertion (iv) is immediate from the deﬁnitions [cf. also assertions (i), (ii); Proposition 5.3]. Finally, we conclude the theory of the present §5 by discussing a certain issue which is closely related to the issue of being of model type. Namely, instead of working at the level of the entire category C, or C Fr-tr , we consider the issue of being “of model type” at the level of a single Frobenius-trivial object: Proposition 5.6. (Base-Sections of Frobenius-Trivial Objects) Suppose that C is of model [hence, in particular, isotropic — cf. Deﬁnition 2.7, (iii)] and unit-proﬁnite type. Let (P, F ) be a base-Frobenius pair of C; A ∈ Ob(P) a def Frobenius-trivial object; A D = Base(A). Then the pair σ : Aut D (A D ) → Aut C (A), φ : N ≥1 → End C (A) — where σ is a group homomorphism whose composite with the natural surjec- tion Aut C (A) Aut D (A D ) [cf. Theorem 5.1, (iii)] is the identity, and φ is a homomorphism of monoids — determined by “restricting” P, F to A, in fact, de- pends only on the data (C, A), and, in particular, is independent of the data (F , P) — up to conjugation [as a pair!] by an element of O × (A). We shall refer to such a pair (σ, φ) as a base-Frobenius pair of A; when F is regarded as being 106 SHINICHI MOCHIZUKI known only up to composition with automorphisms of the monoid N ≥1 , we shall refer to such a pair as a quasi-base-Frobenius pair of A. Proof. Let σ : Aut D (A D ) → Aut C (A), φ : N ≥1 → End C (A) be another such pair, that arises from a base-Frobenius pair (P , F ) of C. Write def def E ⊆ End C (A) for the submonoid of base-isomorphisms; φ n = φ(n) ∈ E, φ n = φ (n) ∈ E, for n ∈ N ≥1 . Then I claim that it suﬃces to show the existence of a u ∈ O × (A) ⊆ E such that u · φ p · u −1 = φ p def def for all p ∈ Primes. Indeed, if, for α ∈ Aut D (A D ), we write σ α = σ(α), σ α = σ (α) — so σ α = v α · u · σ α · u −1 , for some v α ∈ O × (A) ⊆ E — then it follows from the functoriality of F , F that, for p ∈ Primes, σ α · φ p = φ p · σ α ; σ α · φ p = φ p · σ α — hence [since C, being of model type, is also of [birationally] Frobenius-normalized type — cf. Deﬁnition 4.5, (i)] that u · v α ·φ p · σ α · u −1 = v α · u · σ α · φ p · u −1 = v α · (u · σ α · φ p · u −1 ) = v α · (u · σ α · u −1 ) · (u · φ p · u −1 ) = σ α · φ p = φ p · σ α = (u · φ p · u −1 ) · v α · (u · σ α · u −1 ) = (u · v α p · φ p · u −1 ) · (u · σ α · u −1 ) = u · v α p · φ p · σ α · u −1 — which [by the total epimorphicity of C] implies that v α = v α p , for all p ∈ Primes. Thus, by taking p = 2, we obtain that v α = 1. Since φ, φ are homomorphisms, and N ≥1 is generated by Primes, this completes the proof of the claim. To verify the existence of a u ∈ O × (A) as in the above claim, let us ﬁrst observe that if M ⊆ O × (A) ⊆ E is any subgroup such that for any m ∈ M , f ∈ E, there exists an m ∈ M such that f · m = m · f , then there is a natural monoid def structure on the set of cosets E M = M \E = {M · f } f ∈E , together with a natural surjection of monoids E E M . For p ∈ Primes, let us write M p ⊆ O × (A) for the closed subgroup topologically generated by the pro-l portions (O × (A))[l] [cf. Deﬁnition 2.8, (ii)] of O × (A), as l ranges over the primes = p. Note that since the Frobenioid C Fr-tr is of Aut-ample type [cf. Theorem 5.1, (iii)], it follows that any f ∈ E admits a factorization f = f 0 · f 1 , where f 0 is an automorphism, and f 1 is a base-identity endomorphism. Thus, [by applying, again, the fact that C, being of model type, is also of [birationally] Frobenius-normalized type — cf. Deﬁnition 4.5, (i)] it follows that “for any m ∈ M p , there exists an m ∈ M p such THE GEOMETRY OF FROBENIOIDS I 107 that f · m = m · f ”. In particular, it makes sense to speak of the monoid E M p . Let p us use the symbol “ ≈ ” to denote the equality of the images in E p of elements of E. Now since we have a natural isomorphism ∼ O × (A)[p] → O × (A) p∈Primes [cf. Deﬁnition 2.8, (ii)], it thus follows that to prove the existence of a u ∈ O × (A) as desired, it suﬃces to show, for each p ∈ Primes, the existence of a u p ∈ O × (A)[p] p ≈ φ l , for all l ∈ Primes [i.e., we then take u to be the “inﬁnite such that u p ·φ l ·u −1 p product” of the u p ]. p Now observe that for each l ∈ Primes, φ l ≈ v l · φ l , for some v l ∈ O × (A)[p]. p Since, for w ∈ O × (A)[p], we have, for l ∈ Primes, w · φ l · w −1 ≈ w 1−l · φ l [where we recall again that C, being of model type, is also of [birationally] Frobenius-normalized type — cf. Deﬁnition 4.5, (i)], and O × (A)[p] is a [topologically ﬁnitely generated] p ≈ φ p , pro-p group, it follows that there exists a u p ∈ O × (A)[p] such that u p ·φ p ·u −1 p p as well as a w l ∈ O × (A)[p] such that w l · u p · φ l · u −1 ≈ φ l , for each l ∈ Primes. p On the other hand, since φ, φ are homomorphisms, it follows that p φ l 1 · φ l 2 ≈ φ l 2 · φ l 1 ; p φ l 1 · φ l 2 ≈ φ l 2 · φ l 1 [for l 1 , l 2 ∈ Primes]. Thus, for l ∈ Primes, we have p p −1 ≈ w l · u p · φ l · φ p · u −1 ≈ w l · u p · φ l · u −1 w l · u p ·φ p · φ l · u −1 p p p · u p · φ p · u p p p p −1 ≈ φ l · φ p ≈ φ p · φ l ≈ u p · φ p · u −1 p · w l · u p · φ l · u p p p −1 ≈ u p · w l p · φ p · u −1 ≈ w l p · u p · φ p · φ l · u −1 p · u p · φ l · u p p p — which [by the total epimorphicity of C] implies that w l ≈ w l p [for all l ∈ Primes]. Since O × (A)[p] is a [topologically ﬁnitely generated] pro-p group, we thus conclude p that w l ≈ 1. This completes the proof of the existence of a u ∈ O × (A) as desired, and hence of Proposition 5.6. Remark 5.6.1. The notion of a “base-section of a Frobenius-trivial object” [i.e., in the notation of Proposition 5.6, a section “σ”] is intended to be an abstract category-theoretic translation of the notion of a “tautological section of a trivial line bundle” [cf. Remark 2.7.1; the Frobenioids of Examples 6.1, 6.3 below]. Corollary 5.7. (Category-theoreticity of Base-Sections) For i = 1, 2, let Φ i be a perf-factorial divisorial monoid on a connected, totally epimorphic category D i which is Div-slim [with respect to Φ i ]; C i → F Φ i a Frobenioid of standard type; ∼ Ψ : C 1 → C 2 108 SHINICHI MOCHIZUKI an equivalence of categories. If C 1 , C 2 are of group-like type, then we also assume that both Ψ and some quasi-inverse to Ψ preserve base-isomorphisms. Then: (i) Ψ maps base-sections (respectively, quasi-base-Frobenius pairs) of C 1 to base-sections (respectively, quasi-base-Frobenius pairs) of C 2 . In particular, C 1 is of model type if and only if C 2 is. (ii) C 1 is of unit-proﬁnite type if and only if C 2 is. (iii) Suppose that C 1 , C 2 are of model and unit-proﬁnite type. Then Ψ maps every quasi-base-Frobenius pair of a Frobenius-trivial object A 1 ∈ Ob(C 1 ) to a quasi-base-Frobenius pair of a Frobenius-trivial object A 2 ∈ Ob(C 2 ). (iv) Suppose, moreover, when C 1 , C 2 are of group-like type, that both Ψ and some quasi-inverse to Ψ preserve Frobenius degrees. Then the preﬁx “quasi-” may be removed from the statements of (i), (iii). Proof. Indeed, sorting through the deﬁnitions, to verify assertions (i), (ii), (iii), (iv) it suﬃces to show that Ψ preserves isotropic objects, prime-Frobenius morphisms, pull-back morphisms, birationalizations, the natural projection functor C i → D i [hence, in particular, the units “O × (−)”], and [in the case of the ﬁnal portion of assertion (iv), when C 1 , C 2 are not of group-like type] Frobenius degrees. But this follows from Theorem 3.4, (i), (iii); Corollary 4.10; Corollary 4.11, (ii) [cf. also Remark 3.4.1]. This completes the proof of Corollary 5.7. THE GEOMETRY OF FROBENIOIDS I 109 Section 6: Some Motivating Examples In the present §6, we discuss some of the principal motivating examples from arithmetic geometry of the theory of Frobenioids. In particular, in the case of number ﬁelds, one of these examples provides an interesting “category-theoretic interpretation” of some results of classical number theory, such as the Dirichlet unit theorem and Tchebotarev’s density theorem, as well as a result in transcendence theory due to Lang [cf. Theorem 6.4, (i), (iii), (iv)]. Example 6.1. A Frobenioid of Geometric Origin. Let V be a proper normal [geometrically integral] variety over a ﬁeld k; K the function ﬁeld of V ; def D K a set of Q-Cartier K/K a [possibly inﬁnite] Galois extension; G = Gal( K/K); prime divisors on V . The connected objects of the Galois category B(G) [cf. §0] may is a ﬁnite [necessarily separable] be thought of as schemes Spec(L), where L ⊆ K extension of K. If we write V [L] for the normalization of V in L [so V [L] is also a proper normal variety], then let us write D L for the set of prime divisors of V [L] that map into [possibly subvarieties of codimension ≥ 1 of] prime divisors of D K . If for every Spec(L) ∈ Ob(B(G) 0 ) [cf. §0], every prime divisor of D L is Q-Cartier, In the following, we shall assume that then we shall say that D K is K-Q-Cartier. D K is K-Q-Cartier. If L ⊆ K is a ﬁnite extension, then let us write Φ(L) ⊆ Z ≥0 [D L ] ⊆ Z[D L ] for the monoid of Cartier eﬀective divisors D on V [L] with support in D L [i.e., D such that every prime divisor in the support of D belongs to D L ] and B(L) ⊆ L × for the group of rational functions f on V [L] such that every prime divisor at which f has a zero or a pole belongs to D L . Observe that Φ(L) gp ⊆ Z[D L ] may be identiﬁed with the group of Cartier divisors on V [L], and that Φ(L) pf = Q ≥0 [D L ] ⊆ Q[D L ] = (Φ(L) pf ) gp [since D K is K-Q-Cartier]; moreover, one veriﬁes immediately that Φ(L) is perf- ∼ factorial, that there is a natural bijection Prime(Φ(L)) → D L , and that the supports of elements of Φ(L) are precisely the ﬁnite subsets of D L . By assigning to a rational function f the divisor obtained by subtracting the divisor of poles of f from the divisor of zeroes of f , we obtain a natural homomorphism B(L) → Φ(L) gp which is functorial in L. In particular, the assignments L → Φ(L), L → B(L) def determine, respectively, a perf-factorial divisorial monoid Φ on D = B(G) 0 and a group-like monoid B on D, equipped with a homomorphism [of monoids on D] B → Φ gp . Thus, by Theorem 5.2, (ii), this data determines a [model] Frobenioid C V, K,D K 110 SHINICHI MOCHIZUKI of isotropic and birationally Frobenius-normalized type. Note that an object of C V, K,D K that projects to Spec(L) ∈ Ob(B(G) 0 ) may be thought of as a line bundle L on V [L] that is representable by a Cartier divisor D with support in D L . If L is such a line bundle on V [L], and M is such a line bundle on V [M ] [where M ⊆ K is a ﬁnite extension of K], then one veriﬁes immediately that a morphism L → M in C V, K,D K may be thought of as consisting of the following data: (a) a morphism Spec(L) → Spec(M ) over Spec(K) [which thus induces a morphism V [L] → V [M ] over V ]; (b) an element d ∈ N ≥1 ; (c) a morphism of line bundles L ⊗d → M| V [L] on V [L] whose zero locus is a Cartier divisor supported in D L . Also, we observe that [since V [L] is a proper normal variety] for A ∈ Ob(C V, K,D K ) that projects to Spec(L) ∈ Ob(B(G) 0 ), we have × O × (A) = O (A) = k L where k L denotes the algebraic closure of k in L [so k L is a ﬁnite separable extension of k]. Theorem 6.2. (Geometric Frobenioids) For i = 1, 2, let V i be a proper normal [geometrically integral] variety over a ﬁeld k i ; K i the function ﬁeld of V i ; i /K i ); D i def i /K i a [possibly inﬁnite] Galois extension; G i def = Gal( K = B(G i ) 0 ; K i -Q-Cartier set of prime divisors on V i . For Spec(L i ) ∈ Ob(D i ), D K i = ∅ a K write V i [L i ] for the normalization of V i in L i ; D L i for the set of prime divisors of V i [L i ] that map into [possibly subvarieties of codimension ≥ 1 of ] prime divisors of D K i ; Φ i (L i ) ⊆ Z ≥0 [D L i ] ⊆ Z[D L i ] for the monoid of Cartier eﬀective divisors on V i [L i ] with support in D L i ; B i (L i ) ⊆ L × i for the group of rational functions on V i [L i ] whose zeroes and poles are supported on D L i ; B i (L i ) → Φ i (L i ) gp for the natural map; C i for the associated model Frobenioid of Theorem 5.2, (ii). Then: (i) Let ψ : V 2 → V 1 be a dominant morphism of schemes such that the following conditions are sat- isﬁed: (a) D K 2 is equal to the set of prime divisors of V 2 that map into a prime divisor of D K 1 ; (b) the resulting inclusion of function ﬁelds K 1 → K 2 satisﬁes the 2 factors through K 1 ; (c) K 1 condition that the composite inclusion K 1 → K 2 → K is separably closed in K 2 . Then ψ induces a functor Ψ : C 1 → C 2 THE GEOMETRY OF FROBENIOIDS I 111 [well-deﬁned up to isomorphism] that is compatible with Frobenius degrees, the functor D 1 → D 2 induced by the inclusion of ﬁelds K 1 → K 2 , and the natural transformations Φ 1 → Φ 2 | D 1 , B 1 → B 2 | D 1 induced by pulling back divisors and rational functions, respectively, via ψ. (ii) Assume that the data labeled by the index “1” is equal to the data labeled by the index “2” [so in the following, we shall omit these indices]. Also, let us suppose that k is of positive characteristic p. Then the Frobenius morphism ψ : V → V satisﬁes the conditions of (i), hence determines a functor Ψ: C→C which is isomorphic to the naive Frobenius functor [of degree p on C] of Propo- sition 2.1. (iii) We maintain the assumption of (ii) concerning indices. Then the Frobe- nioid C is of isotropic, standard, and birationally Frobenius-normalized of type, but not of group-like type. If, moreover, for every ﬁnite extension L ⊆ K K, and every D ∈ D L , it holds that D lies in the support of the image in Φ(L) gp of an element of B(L), then C is of rationally standard type. (iv) We maintain the assumption of (ii) concerning indices. Then D is Frobe- nius-slim. Let Z ⊆ G be the subgroup of elements that commute with some open subgroup of G. Then D is slim if and only if Z = {1}; D is Div-slim [relative to Φ] if and only if, for every 1 = z ∈ Z, there exists a ﬁnite Galois extension L ⊆ K of K such that z acts nontrivially on Φ(L). Proof. First, we consider assertion (i). Note that by assumptions (b), (c) [in 1 of K 1 the statement of assertion (i)], it follows that any ﬁnite extension L 1 ⊆ K def 2 of K 2 such that [L 2 : K 2 ] = [L 1 : determines a ﬁnite extension L 2 = L 1 · K 2 ⊆ K K 1 ]. Thus, ψ determines a functor D 1 → D 2 . Moreover, by assumption (a) [in the statement of assertion (i)], it follows that by pulling back [Cartier] divisors and rational functions via ψ, we obtain compatible natural transformations Φ 1 → Φ 2 | D 1 , B 1 → B 2 | D 1 . Thus, it follows formally from the deﬁnition of the category underlying a model Frobenioid in Theorem 5.2, (i), that we obtain a functor Ψ : C 1 → C 2 satisfying the properties stated in assertion (i). From this deﬁnition of the functor Ψ, it then follows immediately from the deﬁnition of the “Frobenius morphism in characteristic p”, together with the deﬁnition of the “naive Frobenius functor” of Proposition 2.1 — i.e., in a word, that both functors are obtained by “raising to the p-th power” — that these two functors are isomorphic. This completes the proof of assertions (i), (ii). Next, we consider assertion (iii). The fact that C is of isotropic and birationally Frobenius-normalized type follows formally from Theorem 5.2, (ii). The fact that C is not of group-like type is immediate from our assumption that D K = ∅ [and the deﬁnition of Φ]. It is immediate that every monomorphism of D is an isomorphism, hence that D is of FSM-type [hence also of FSMFF-type — cf. §0]. If a K-linear of K induces an automorphism of Φ(L) automorphism α of a ﬁnite extension L ⊆ K 112 SHINICHI MOCHIZUKI which preserves the primes of L, then it is immediate from the fact that α induces an automorphism of the scheme V [L] that α maps every prime divisor D ∈ Φ(L) to D [i.e., not to some n · D, where n ≥ 2]; thus, we conclude that Φ is non-dilating, hence that C is of standard type. Now suppose that for every ﬁnite extension L ⊆ K gp of K, and every D ∈ D L , it holds that D lies in the support of the image in Φ(L) of an element of B(L). Then it follows formally [cf. Deﬁnition 4.5, (ii)] that C is of [strictly] rational type [since Φ has already been observed to be perf-factorial — cf. Example 6.1]. Thus, C satisﬁes condition (a) of Deﬁnition 4.5, (iii). Now I claim that every object of (C un-tr ) birat is Frobenius-compact. Indeed, if α is a K-linear of K that acts by multiplication by automorphism of a ﬁnite extension L ⊆ K birat pf λ ∈ Q >0 on Φ (L) ( = 0), then since α induces an automorphism of the variety V [L], it follows that the order ∈ Q >0 of the zero [or pole] of highest order of an element f ∈ Φ birat (L) pf is preserved by α, hence that λ = 1. This completes the proof of the claim, and hence of the fact that C is of rationally standard type. is a ﬁnite Finally, we consider assertion (iv). First, we observe that if L ⊆ K extension of K that corresponds to an open subgroup H ⊆ G, then there is a natural isomorphism ∼ (Z ⊇) Z G (H) → Aut(D Spec(L) → D) [cf. [Mzk7], Corollary 1.1.6]. Since G is proﬁnite, hence, in particular, residually ﬁnite, it follows formally that Z, Z G (H) are also residually ﬁnite, hence that D is Frobenius-slim, by Remark 3.1.2. Also, since Z is the union of subgroups of G of the form “Z G (H)”, it follows formally that D is slim if and only if Z = {1}, and that D is Div-slim [relative to Φ] if and only if, for every 1 = z ∈ Z, there exists of K such that z acts nontrivially on Φ(L ). This a ﬁnite Galois extension L ⊆ K completes the proof of assertion (iv). Remark 6.2.1. Theorem 6.2, (ii), constitutes the principal justiﬁcation for the name “Frobenius functor” that was applied to various functors in §2. From this point of view, the decomposition of the naive Frobenius functor of Proposition 2.1 into “unit-linear” and “unit-wise” Frobenius functors [cf. the proof of Corollary 2.6] may be thought of as corresponding to the decomposition of the Frobenius morphism in positive characteristic algebraic geometry over a ﬁxed base into the composite of a “relative Frobenius morphism”, which is linear over the ﬁxed base, with the Frobenius morphism of the ﬁxed base. Example 6.3. A Frobenioid of Arithmetic Origin. Let F be a number ﬁeld [cf. §0]. Write V(F ) for the set of valuations on F [where we identify com- plex archimedean valuations with their complex conjugates]; O F for the ring of integers of F . If v ∈ V(F ), then we shall write F v for the completion of F at v; O v × ⊆ F v × for the group of units [i.e., elements of valuation 1 of F v × ]; O v ⊆ F v × for the multiplicative monoid of elements of valuation ≤ 1; μ(F ) ⊆ O F × for the def def group of roots of unity in F ; ord(F v ) = F v × /O v × ; ord(O v ) = O v /O v × ⊆ ord(F v ). Thus, ord(F v ) = ord(O v ) gp ; ord(F v ) ∼ = Z, ord(O v ) ∼ = Z ≥0 if v is nonarchimedean; THE GEOMETRY OF FROBENIOIDS I 113 ord(F v ) ∼ = R, ord(O v ) ∼ = R ≥0 if v is archimedean. We shall refer to an element of the monoid def ord(O v ) Φ(F ) = v∈V(F ) as an eﬀective arithmetic divisor on F , and to an element of the group Φ(F ) gp = ord(F v ) v∈V(F ) as an arithmetic divisor on F . Thus, there is a natural homomorphism of groups def B(F ) = F × → Φ(F ) gp [given by mapping an element f ∈ F × to the images of f in the various factors F v × /O v × = ord(F v ), all but a ﬁnite number of which are zero]. Note, moreover, that Φ, B, as well as the homomorphism B → Φ gp are functorial in the number ﬁeld def F . Thus, if F is a [not necessarily ﬁnite] Galois extension of F , G = Gal( F /F ), def D = B(G) 0 , then Φ, B determine monoids on D, and we have a natural homo- morphism B → Φ gp . Moreover, for each ﬁnite extension L ⊆ F of F , one veri- ﬁes immediately that Φ(L) = 0 is perf-factorial, that there is a natural bijection ∼ Prime(Φ(L)) → V(L), and that the supports of elements of Φ(L) are precisely the ﬁnite subsets of V(L). Thus, by Theorem 5.2, (ii), this data determines a [model] Frobenioid C F /F of isotropic and birationally Frobenius-normalized type. Note that an object of C F /F that projects to Spec(L) ∈ Ob(B(G) 0 ) may be thought of as an arithmetic line bundle L on L [i.e., a line bundle on Spec(O L ), equipped with Hermitian metrics at the archimedean primes — cf. [Szp], pp. 13-14]. If L is an arithmetic line bundle on L, and M is an arithmetic line bundle on M [where M ⊆ F is a ﬁnite extension of F ], then one veriﬁes immediately that a morphism L → M in C F /F may be thought of as consisting of the following data: (a) a morphism Spec(L) → Spec(M ) over Spec(F ); (b) an element d ∈ N ≥1 ; (c) a nonzero morphism of arithmetic line bundles L ⊗d → M| L on L. Also, we observe that for A ∈ Ob(C F /F ) that projects to Spec(L) ∈ Ob(B(G) 0 ), we have O × (A) = O (A) = μ(L) [cf., for instance, [Szp], p. 15]. Also, observe that we have a natural arithmetic degree homomorphism : Φ(L) gp → R deg arith L obtained as follows: If v is archimedean, so we have a natural embedding of topo- logical ﬁelds R → F v , then the restriction of deg arith to the factor ord(F v ) maps L the image of λ ∈ R >0 to −[F v : R] · log(λ). If v is nonarchimedean, then the re- striction of deg arith to the factor ord(F v ) maps the image of an element λ ∈ O v to L 114 SHINICHI MOCHIZUKI the natural logarithm of the cardinality of the ﬁnite set O v /(λ) [where O v is the vanishes on the ring of integers of F v ]. Thus, one veriﬁes immediately that deg arith L image of B(L). Remark 6.3.1. In light of Examples 6.1, 6.3, many readers might expect that the next natural step is to attempt to apply the theory of Frobenioids to study arith- metic line bundles on higher-dimensional arithmetic varieties. This leads, however, to numerous complications which are beyond the scope of the present paper. More- over, it is not even clear to the author at the time of writing that this constitutes a natural direction in which to further develop the theory of Frobenioids. Theorem 6.4. (Arithmetic Frobenioids) For i = 1, 2, let F i be a number def def ﬁeld; F i /F i a [possibly inﬁnite] Galois extension; G i = Gal( F i /F i ); D i = B(G i ) 0 ; Φ i the monoid on D i given by the eﬀective arithmetic divisors; B i the group-like monoid on D i given by the multiplicative group of the ﬁeld extension of F i in question; B i → Φ gp i the natural map; C i the associated model Frobenioid of Theorem 5.2, (ii). Then: (i) Assume that the data labeled by the index “1” is equal to the data labeled by the index “2” [so in the remainder of the present assertion (i), we shall omit these indices]. Then the Frobenioids C, C pf , C rlf , C un-tr , (C pf ) un-tr are of isotropic and rationally standard type, but not of group-like type; D is Frobenius-slim and Div-slim [with respect to Φ, Φ pf , Φ rlf ]. Moreover, D is slim if and only if the subgroup of elements of G that commute with some open subgroup of G is trivial. Finally, if A ∈ Ob(C rlf ) is a Frobenius-trivial object that projects to the object of D determined by a ﬁnite extension L ⊆ F of F , then deg arith determines an L isomorphism of groups ∼ δ A : Pic Φ (A) → R [cf. Theorem 5.1, (i)]. (ii) Let ∼ Ψ rlf : C 1 rlf → C 2 rlf be an equivalence of categories between the realiﬁcations [cf. Proposition 5.3] of C 1 , C 2 . Then there exists an element deg(Ψ rlf ) ∈ R >0 such that for all Frobenius-trivial A 1 ∈ Ob(C 1 ), A 2 ∈ Ob(C 2 ) such that A 2 = Ψ rlf (A 1 ) [where we recall that Ψ rlf preserves Frobenius-trivial objects — cf. (i); Corollary 4.11, (iv)], ∼ the composite of δ A 2 with the isomorphism Pic Φ (A 1 ) → Pic Φ (A 2 ) determined by Ψ rlf [cf. (i) above; Corollary 4.10; Corollary 4.11, (iii)] is equal to deg(Ψ rlf ) · δ A 1 . (iii) If the equivalence of categories Ψ rlf of (ii) arises from an equivalence of categories ∼ (Ψ pf ) un-tr : (C 1 pf ) un-tr → (C 2 pf ) un-tr THE GEOMETRY OF FROBENIOIDS I 115 between the unit-trivialized perfections of C 1 , C 2 [cf. (i); Corollary 5.4], then deg(Ψ rlf ) ∈ Q >0 . In particular, if A 1 ∈ Ob((C 1 pf ) un-tr ) [whose projection to D 1 we denote by Spec(L 1 )], A 2 ∈ Ob((C 2 pf ) un-tr ) [whose projection to D 2 we denote by Spec(L 2 )], A 2 = (Ψ pf ) un-tr (A 1 ), then the bijection ∼ ∼ ∼ V(L 1 ) → Prime(Φ 1 (L 1 )) → Prime(Φ 2 (L 2 )) → V(L 2 ) induced by (Ψ pf ) un-tr [cf. (i); Corollary 4.11, (iii)] maps a valuation v 1 ∈ V(L 1 ) lying over a valuation v 0 of Q to a valuation v 2 ∈ V(L 2 ) lying over the valuation v 0 of Q. (iv) If the equivalence of categories Ψ rlf of (ii) arises from an equivalence of categories ∼ Ψ : C 1 → C 2 between C 1 , C 2 [cf. (i); (iii); Theorem 3.4, (iii), (iv)], then deg(Ψ rlf ) = 1. If, moreover, there exists a ﬁnite extension L 1 ⊆ F 1 of F 1 which is Galois over Q, ∼ then the corresponding [i.e., via the equivalence D 1 → D 2 induced by Ψ — cf. (i); Corollary 4.11, (ii)] ﬁnite extension L 2 ⊆ F 2 of F 2 is isomorphic to L 1 in a fashion that is compatible with an isomorphism F 1 ∼ = F 2 . Proof. First, we consider assertion (i). We have already seen in Example 6.3 that the Frobenioid C is of isotropic and birationally Frobenius-normalized type, and that Φ is nonzero [so C is not of group-like type] and perf-factorial. As was observed in the proof of Theorem 6.2, (iii), (iv), D is Frobenius-slim and of FSM-type, hence also of FSMFF-type. Moreover, since any automorphism of a number ﬁeld that ﬁxes all of the valuations of the number ﬁeld is clearly equal to the identity automorphism, it follows immediately that Φ is non-dilating, and that D is Div-slim [relative to Φ, hence also relative to Φ pf , Φ rlf ]. Also, it is immediate from the deﬁnition of B that C is of [strictly] rational type, and that every object of (C un-tr ) birat is Frobenius- compact. Thus, we conclude that C [hence also C pf , C rlf , C un-tr , (C pf ) un-tr — cf. Proposition 5.5, (iii)] is of rationally standard type. The proof of the criterion for D to be slim is entirely similar to the proof given for Theorem 6.2, (iv). Finally, to show that the surjection δ A : Pic Φ (A) R is, in fact, an isomorphism, it suﬃces to verify that the image of Φ birat (L) ⊗ Z R = gp rlf gp (L × )⊗ Z R in (Φ rlf factor ) (L) is equal to the set of elements of (Φ factor ) (L) with ﬁnite support whose image under deg arith is 0. But this is an immediate consequence of L the well-known Dirichlet unit theorem of classical number theory [cf., e.g., [Lang2], p. 104]. This completes the proof of assertion (i). Now assertion (ii) follows by observing that the isomorphism of groups ∼ Pic Φ (A 1 ) → Pic Φ (A 2 ) determined by Ψ rlf [cf. assertion (i); Corollary 4.10; Corollary 4.11, (iii)] is compat- ible with the “order structure” induced on both sides [via δ A 1 , δ A 2 ] by the “order 116 SHINICHI MOCHIZUKI structure” of R. [Indeed, this compatibility follows from the fact that the isomor- ∼ rlf phism in question arises from an isomorphism of monoids Φ rlf 1 (A 1 ) → Φ 2 (A 2 ).] This completes the proof of assertion (ii). Next, we observe that assertion (iii) follows formally from assertion (ii), by applying Lemma 6.5, (ii), below in the following fashion: If deg(Ψ rlf ) ∈ Q >0 , then one veriﬁes immediately that there exist three nonarchimedean valuations w 1 , w 3 , w 5 ∈ V(L 1 ) lying over primes p 1 , p 3 , p 5 ∈ Primes, respectively, with the property that w 1 → w 2 ∈ V(L 2 ), w 3 → w 4 ∈ V(L 2 ), w 5 → w 6 ∈ V(L 2 ), where w 2 , w 4 , w 6 lie over primes p 2 , p 4 , p 6 ∈ Primes, respectively, such that p 1 , p 2 , p 3 , p 4 , p 5 , p 6 are distinct. But this implies that log(p 1 )/log(p 2 ), log(p 3 )/log(p 4 ), log(p 5 )/log(p 6 ) ∈ (deg(Ψ rlf )) −1 · Q >0 in contradiction to Lemma 6.5, (ii). Thus, deg(Ψ rlf ) ∈ Q >0 . The ﬁnal portion of assertion (iii) concerning valuations of Q now follows from Lemma 6.5, (i). This completes the proof of assertion (iii). Finally, we consider assertion (iv). Suppose that v 1 ∈ V(L 1 ) maps to v 2 ∈ V(L 2 ) [cf. the notation of the statement of assertion (iii)]. For i = 1, 2, write deg(L i , v i ) for the number of valuations ∈ V(L i ), including v i , that lie over the same valuation of Q as v i . Then by Tchebotarev’s density theorem [cf., e.g., [Lang2], Chapter VIII, §4, Theorem 10], it follows that [L i : Q] is equal to the maximum of the deg(L i , v i ), as v i ranges over the elements of V(L i ). Moreover, if v i is nonarchimedean and lies over a prime p i ∈ V(L i ), then p i splits completely in L i if and only if deg(L i , v i ) = [L i : Q]. Thus, it follows that if v 1 , v 2 lie over a prime p ∈ Primes [cf. assertion (iii)], then [again by assertion (iii)] p splits completely in L 1 if and only if p splits completely in L 2 . If this is the case, then it follows that deg arith maps a generator L i of the monoid Φ i (L i ) v i ( ∼ Z ) to log(p). Thus, we conclude that deg(Φ rlf ) = 1, as = ≥0 desired. Note that this implies that v 1 is of degree 1 [i.e., deg arith maps a generator L 1 ∼ of the monoid Φ 1 (L 1 ) v 1 ( = Z ≥0 ) to log(p)] if and only if v 2 is of degree 1. Thus, if L 1 is Galois over Q, then whenever v 2 is of degree 1, it follows that v 1 is of degree 1, hence that p splits completely in L 1 [since L 1 is Galois over Q]. But this implies [again by Tchebotarev’s density theorem — cf., e.g., [NSW], Theorem 12.2.5] that L 1 ⊆ L 2 , hence that L 1 = L 2 [since we have already seen that [L 1 : Q] = [L 2 : Q]]. This completes the proof of assertion (iv). Lemma 6.5. Numbers) (Transcendental Properties of Logarithms of Prime (i) The real numbers log(p) ∈ R, where p ranges over the prime numbers, are linearly independent over Q. (ii) Let p 1 , p 2 , . . . , p 6 be distinct prime numbers. Then there do not exist λ 1 , λ 2 ∈ Q >0 such that: log(p 1 )/log(p 2 ) = λ 1 ·log(p 3 )/log(p 4 ) = λ 2 ·log(p 5 )/log(p 6 ). THE GEOMETRY OF FROBENIOIDS I 117 Proof. Assertion (i) is a formal consequence of the fact that Z is a unique factor- ization domain. Assertion (ii) is a consequence of a theorem of Lang [cf. [Lang1]; [Baker], p. 119]: Indeed, since the log(p i ) are linearly independent over Q [by as- sertion (i)], it follows that each of the following two sets of numbers is also linearly independent over Q: {log(p 2 ), log(p 4 ), log(p 6 )}; {1, log(p 3 )/log(p 4 )} Moreover, all six products of one element from the ﬁrst set and one element from the second set are of the form μ · log(p i ), where μ ∈ Q >0 . Thus, the exponential of each of these products is algebraic, in contradiction to Lang’s theorem. 118 SHINICHI MOCHIZUKI Appendix: Slim Exponentiation In the present Appendix, we discuss some elementary general nonsense con- cerning slim categories. Deﬁnition A.1. (i) A 2-category of 1-categories will be called 2-slim [cf. [Mzk7], Deﬁnition 1.2.4, (iii)] if every 1-morphism [i.e., functor] in the 2-category has no nontrivial automorphisms. (ii) If D is a 2-category of 1-categories, then we shall write |D| for the associated 1-category whose objects are objects of D and whose morphisms are isomorphism classes of morphisms of D [cf. [Mzk7], Deﬁnition 1.2.4, (iv)]. We shall also refer to |D| as the coarsiﬁcation of C. Remark A.1.1. The name “coarsiﬁcation” is motivated by the theory of “coarse moduli spaces” associated to (say) “ﬁne moduli stacks” [cf. [Mzk7], Remark 1.2.4.1]. The following result may be regarded as a generalization of [Mzk7], Proposition 1.2.5, (ii) [a result concerning anabelioids], to the case of arbitrary slim categories. Proposition A.2. (Slim Exponentiation) Let C be a slim category [cf. §0]. Let D be the 2-category of 1-categories deﬁned as follows: The objects of D are the categories C A [cf. §0], where A ∈ Ob(A). The 1-morphisms of D are the functors f ! : C A → C B [cf. §0] induced by morphisms f : A → B of C. The 2-morphisms of D are isomorphisms between these functors [cf. §0]. Then D is 2-slim. Moreover, the functor E : C → |D| A → C A ; f → f ! ∼ determines an equivalence of categories C → |D|. We shall refer to the functor E as the slim exponentiation functor. Proof. The fact that D is 2-slim follows immediately from the assumption that C is slim. Now it is immediate from the deﬁnitions that E is full and essentially surjective. To verify that E is faithful, let us ﬁrst observe that given any two morphisms f, g : A → B of C, an isomorphism between the functors f ! , g ! : C A → C B determines an isomorphism between the composites of the functors f ! , g ! with THE GEOMETRY OF FROBENIOIDS I 119 the natural functor C B → C. On the other hand, these two composite functors C A → C both coincide with the natural functor C A → C [i.e., that maps an object ∼ C → A of C A to the object C of C]. Thus, any isomorphism f ! → g ! determines an automorphism of the natural functor C A → C, which [by the slimness of C!] is the ∼ identity automorphism. But this implies [by applying the isomorphism f ! → g ! to id A the object A −→A of C A ] that f = g, as desired. 120 SHINICHI MOCHIZUKI Index 1-commutative, §0 2-slim, A.1, (i) abstract equivalence, §0 almost totally epimorphic, §0 anchor, §0 A-pair, proof of 5.1, (i) arithmetic degree, 6.3 arithmetic divisor, 6.3 arithmetic line bundle, 6.3 Aut-ample, 1.2, (iv), (v) Aut-saturated, §0 Aut sub -ample, 1.2, (iv), (v) Aut sub -saturated, §0 Aut-type, §0 base category, 1.1, (iii), (iv) base-equivalent, 1.2, (ii) base-Frobenius pair (of a Frobenioid), 2.7, (iii) base-Frobenius pair (of a Frobenius-trivial object), 5.6 base-FSM-morphism, 1.2, (ii) base-identity, 1.2, (ii) base-isomorphism (base-isomorphic), 1.2, (ii) base-section, 2.7, (i) base-trivial, 1.2, (iv), (v) birationalization (of a Frobenioid), 4.4 birationally Frobenius-normalized, 4.5, (i) bounded, §0 categorical ﬁber product, §0 categorical quotient, §0 centralizer, §0 characteristic, §0 characteristically injective, §0 characteristic splitting, 2.3 characteristic type, §0 co-angular, 1.2, (iii) coarsiﬁcation, A.1, (ii) connected category, §0 co-objective, §0 co-primary, 4.1, (iii) co-prime type, 2.8, (iii) Div-equivalent, 1.2, (ii) Div-Frobenius-trivial, 1.2, (iv), (v) Div-identity, 1.2, (ii) THE GEOMETRY OF FROBENIOIDS I divisorial, 1.1, (i) divisor monoid, 1.1, (iv); 5.2, (ii) Div-slim, 4.5, (iv) eﬀective arithmetic divisor, 6.3 elementary Frobenioid, 1.1, (iii) End-ample, 1.2, (iv), (v) End-equivalence, §0 factorization homomorphism, 2.4, (i), (c) factorization of morphisms of a Frobenioid, 1.3, (iv), (a) factorization of pre-steps of a Frobenioid, 1.3, (v), (b), (c) F -distinguished, 2.7, (ii) ﬁberwise surjective, §0 ﬁnitely (respectively, countably) connected type, §0 F P-path, proof of 5.2, (iv) Frobenioid, 1.3 Frobenius-ample, 1.2, (iv), (v) Frobenius-compact, 1.2, (iv), (v) Frobenius degree, 1.1, (iii), (iv) Frobenius functor (on an elementary Frobenioid), 2.4, (iii) Frobenius-isotropic, 1.2, (iv), (v) Frobenius-normalized, 1.2, (iv), (v) Frobenius-section, 2.7, (ii) Frobenius-slim, 3.1, (i) Frobenius-trivial, 1.2, (iv), (v) FSMFF-type (category of), §0 FSMI-morphism, §0 FSM-morphism, §0 FSM-type (category of), §0 groupiﬁcation, §0 group-like (monoid), 1.1, (i) group-like (object of a pre-Frobenioid), 1.2, (iv), (v) immobile, §0 integral, §0 irreducible (element of a monoid), §0 irreducible (morphism of a category), §0 isometric morphism (isometry), 1.2, (i) iso-subanchor, §0 isotropic, 1.2, (iv), (v) isotropic hull, 1.2, (iv) isotropiﬁcation functor, 1.9, (v) K-Q-Cartier, 6.1 LB-invertible, 1.2, (iii) left-hand isomorphism, 4.2, (iii) linear morphism, 1.2, (i) 121 122 SHINICHI MOCHIZUKI metrically equivalent, 1.2, (i) metrically trivial, 1.2, (iv), (v) mid-adjoint, §0 minimal-adjoint, §0 minimal-coadjoint, §0 mobile, §0 model Frobenioid, 5.2, (ii) model type, 4.5, (i) monoid, §0 monoid (on a category), 1.1, (ii) monoid type, §0 mono-minimal, §0 monoprime, §0 morphism of Frobenius type, 1.2, (iii) naive Frobenius functor, 2.1, (i) natural projection functor, 1.1, (iii), (iv) non-dilating (endomorphism), 1.1, (i) non-dilating (monoid on a category), 1.1, (ii) number ﬁeld, §0 one-morphism category, §0 one-object category, §0 opposite category, §0 (p 1 , p 2 )-admissible, proof of 3.4 P-distinguished, 2.7, (i) perfection (of a Frobenioid), 3.1, (iii) perfection (of a monoid), §0 perfect (monoid), §0 perfect (object of a pre-Frobenioid), 1.2, (iv) perf-factorial, 2.4, (i) pre-divisorial, 1.1, (i) pre-Frobenioid, 1.1, (iv) pre-Frobenioid structure, 1.1, (iv) pre-model type, 2.7, (iii) pre-step, 1.2, (iii) primary (element of a monoid), §0 primary (pre-step), 1.2, (iii) prime, §0 prime-Frobenius morphism, 1.2, (iii) pro-l portion, 2.8, (ii) pseudo-terminal, §0 pull-back morphism, 1.2, (ii) quasi-base-Frobenius pair (of a Frobenioid), 2.7, (iii) quasi-base-Frobenius pair (of a Frobenius-trivial object), 5.6 quasi-connected, §0 quasi-Frobenius-section, 2.7, (ii) THE GEOMETRY OF FROBENIOIDS I quasi-Frobenius-trivial, 1.2, (iv), (v) quasi-isotropic type, 3.1, (i) raising to the ζ-th power, 2.8, (iii) rational function monoid, 4.4, (ii); 5.2, (ii) rationally standard type, 4.5, (iii) rational object, 4.5, (ii) rational type, 4.5, (ii) realiﬁcation (of a Frobenioid), 5.3 realiﬁcation (of a perf-factorial monoid), 2.4, (i) residually ﬁnite group, 3.1.2 right-hand isomorphism, 4.2, (iii) rigid, §0 saturated, §0 sharp, §0 skeletal subcategory, §0 skeleton, §0 slim (category), §0 slim exponentiation functor, A.2 slim (proﬁnite group), §0 standard Frobenioid, 1.1, (iii) standard type, 3.1, (i) step, 1.2, (iii) strictly rational object, 4.5, (ii) strictly rational type, 4.5, (ii) subanchor, §0 sub-automorphism, §0 subordinate, §0 sub-quasi-Frobenius-trivial, 1.2, (iv), (v) supporting monoid type, 2.4, (ii) support (of an element of a perf-factorial monoid), 2.4, (i), (d) supremum, §0 terminal, §0 totally epimorphic, §0 twin-primary, proof of 4.9 unit-equivalence, 3.1, (iv) unit-linear Frobenius functor, 2.5, (iii) unit-proﬁnite type, 2.8, (i) unit-trivial, 1.2, (iv), (v) unit-trivialization (of a Frobenioid), 3.1, (iv) unit-wise Frobenius functor, 2.6, 2.9 universally Div-Frobenius-trivial, 1.2, (iv), (v) zero divisor, 1.1, (iii), (iv) 123 124 SHINICHI MOCHIZUKI Chart of Types of Morphisms in a Frobenioid type of morphism projection to base zero divisor Frobenius degree linear isometry base-isomorphism base-FSM-morphism pull-back morphism pre-step step primary pre-step isometric pre-step LB-invertible morphism of Frobenius type prime- Frobenius morphism ? ? isomorphism FSM-morphism ? isomorphism isomorphism isomorphism isomorphism ? ? 0 ? ? 0 ? =0 primary 0 0 1 ? ? ? 1 1 1 1 1 ? isomorphism 0 ? isomorphism 0 prime THE GEOMETRY OF FROBENIOIDS I 125 Bibliography [Baker] A. 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